Learning what ?

Very short answer: go with C#.

Short answers:
a. If you learn C# then you didn't get C++ for free.
b. Yes C# is independent of C++.
c. No.
d. Practically, no (porting the whole .NET Framework would be hard).

:)

The shortest possible answer: C++ is very different language than C# except for the similar syntax.

I understand that your priority is to be productive. Then, go with C#. C# is a much cleaner and safer programming language, you get array-index checking, better syntax for generics, etc.

a.) No, you can't write something in C# and then port it to C++. They are simply two different languages. C# looks and feels more like Java than C++.
b.) Yes, C# is independent
c.) no
d.) no

In addition, if you really want to be productive, then learn specific topics in C# and not just the language.
1.) Interoperability with C and C++. Both languages produce either .dll (series of classes or functions without a main entry point) or .exe (.dll with a main entry point). C# can import the dll or exe and call functions. I suggest you learn how to call .dll in the operating system (namely System32, System, etc.). These give you access to all hardware attached to the system and all available services
2.) ADO.Net and database communications. You should learn how to query databases (MSSQL, MYSQL, ORACLE). Get a feel for System.Data (DataTable, DataRow, DataColumn, etc.)
3.) ASP.Net and how to create websites and webservices. Learn how to communicate with facebook, twitter, amazon, etc.
4.) Custom Control Creation. Basically, how to create custom controls out of existing controls. This could be a better datagridview, a better button or I don't know, I group of controls that do something specific *(e.g. login panel)
5.) C# Component Vendors. There are companies that sell ready-made controls such as Dev Express. Their products have trial versions. They are extensions of C# and they make C# look a lot better
6.) VISUAL STUDIO 2005 or higher. This should be at the start. It's almost inconceivable to do C# without Visual Studio. Get used to the project panels (dockable on the left), the palette and the properties window. Simply get used to all the buttons in there as well as the working area. Also know about the refactor tools: Safely rename, extract method, encapsulate fields, etc. They should save you a lot of time in the future.

On the final note, don't be afraid to use the GUI components in C#. Unlike in Java Netbeans where the generated code is mostly uneditable and inefficient, C# generated code are editable two ways (design or source). and the delegate model would make event wiring extremely efficient.

If your goal was to make your learning time productive, I hope this post is helpful. If you like it, please accept it as an answer.

In complement to Roger, let me add some more information.

C++ is an evolution of C towards OOP and Generic programming (this second was added later with the income of templates, and was mainly "discovered by using templates", so the language syntax is a little bit fuzzy) actually introducing some "functional programming" techniques (via "lambda").

It retains the C "original sin" that consider the memory as a resource that must be managed by the programmer. In C this is a requirement (since it is a "high level assembler") But C++ -to retain this- introduces OOP without making a clear distinction between objects and values (you can instantiate a class either on heap and refer to it with a pointer, as well as on the stack or as an embedded member) and hence having to distinguish in syntax between address and reference. With the problem that is no always so clear who has to destroy something, and how can every other be informed of such an event. And that's what makes C++ "difficult".

On the other side, Java removed a number of C++ features (the ones that seemed to its designers "dangerous") and introduced a garbage collected memory model, and the idea that every object is on a managed heap.
This simplify a lot the OOP programming, but eliminates some useful thing certain abstract operation are comfortable with (like operator overloading, default parameter,...).
Java is not properly a "compiled language" mainly because a machine hardware processor don't have a "garbage collected heap". It requires a "virtual machine" dealing at least with the memory handling stuff.
Java designer -by the way- specify the interface between such such a machine and the language so that the language can be compiled in a way that no knowledge is required about the hardware instruction set, thus making Java programming suitable for every Java machine implementation for whatever platform.

Microsoft, to react to expansion of Java, introduced the .Net initiative, that -essentially- made available a "garbage collected heap and a virtual machine speaking MSIL - MS Intermediate Language", that -unlike the JVM- is much more similar in is "instruction set" to the underlying x86 processors. (an MSIL assembly program is compiled just before it is executed into processor instruction and then executed: this makes programs slower in start but faster in execution)
Since C++ (the most used language for COM object development, whose are mostly based many MS component interfaces) cannot interact with such an environment (there is no "relocable pointer" - also known as "handle" - and "tracked reference" for that) an new language (C#) was developed.
Like Java, it introduces a clear distinction between values (that must be on stack or somehow embedded in objects) and objects (that are always on heap) thus making the responsibility of the garbage collector clear, and the need of distinct pointer and reference syntax useless,thus eliminating the most of C++ fuzziness.
Unlike Java, C# retains C++ features like operator overloading, letting "type abstraction" still possible, and low level access still possible with unsafe operation that can still be explicitly required where needed.

Similarly, C++ had also been extended by Microsoft into C++/CLI by adding a number of keywords to let C++ able also to interact with "managed" environment.
As far today (2010) things seems to me, such "extension" are not widely accepted from both the C++ community (Who does more "system programming stuffs" don't like "managed environment") and the .Net community (that prefer the more simple C#, for managed stuffs).

To enforce the use of .Net, Miscorsoft came with a framework that today has a huge class library, making relatively simple every sort of "Programming" either about GUI, Networking, Web, Multimedia etc.
C++ cannot access such libraries and must therefore use natively compiled (or compilable) libraries.
Conversely, C++ compiler are available for virtually whatever processor, while MSIL JIT compiler for processor other than x86 are a rarity. (but x86 means both Windows and Linux, that is 98% of the end-user and data-center OSs of today: mono is -in its essence- the Linux implementation of an MSIL compiler).

Now to come to your questions:

a) Depends: C# is a simplified C++: going to C++ from C# means learn how to manage yourself memory and manage yourself addressing and pointers.
But the most of the GUI programming does not depend on the language but on the library used to manage the graphic interaction.
If your "calculator program" spits results on the "standard output" after taking inputs from the "standard input" ... C++ and C#are different only in what I just told.
If it has a graphic user interface, in C# you will probably use WinForm or WPF (.Net libraries), in C++ you will probably fall-through to the native Win32 C API or to some C++ wrapper like MFC or like WxWidget, Qt etc.
Completely different experiences.

b)yes: they are two well defined languages, each with its own specification. One is not a preprocessor for the other. If I well understood the question.

c)yes, by invoking csc from the command line. But it still requires a .Net framework equipped machine to run (Just like a Win32 Application requires a Win32 "equipped machine" to run: you cannot run a Windows exe on linux, after all!)

d)Ideally yes, until a .Net equivalent MSIL machine exist for that platform. Apart from "mono", I'm not aware of any other implementation on.

In substance, if you want to be a windows developer (including Windows mobile) C# is probably more productive. If you want to be cross platform, Java for OOP and C++ for "native" are more widely available, but less "feature reach".

If you're not interested in immediate feature, but are foreseeing a long term language, give also a look to D (here[^].
It is still immature in terms of frameworks and tools, but as a language it eliminates all the C++ fuzziness taking many C++, C# and Java paradigms (including both RAII and garbage collected OOP) good functional and meta-programming (D templates are far better then C++ ones) and compiles in native machine code. (and can be integrated in MS Visual Studio via visual D[^]
It is gaining popularity in game programming, since it easily interface native C libraries, but is still in its "nerd stage" for certain aspects.

I came from a C++ back ground and picking up C# from it is honestly quite trivial. There are some things that take getting use to and as you oberve them you often are a better programmer than that which took the straight C# route, as they do not see the bennefit of using certain features and pick one of the other ways to skin a cat. Hard to give a concrete example here.

But more importantly we have found that those that learn C# first have huge difficulties in picking up C++. They tend to create numerous bugs or simply can not comprehend how to get a simple data structure working (memory mapping and such).

With that said, my recomendation is learn C++ first. You need not learn in completely in depth but get the basics and some of the intermediate concepts down. After that learn C# continue to study it until you consider your self an expert.

In the end you will be able to support legacy software, native code required apps, and any new concepts.

1) C++ is as similar to Java as C++ is similar to C#. Similar syntax in many cases, but different behavior and significantly different syntax when you really get into things. The frameworks are what is really different (though, technically, you can use the .Net Framework with C++). If you wrote a C# calculator using C#, you'd probably be making use of Windows Forms or WPF and the .Net Framework. In C++, you would likely use lower level libraries, so it would be a very different (and more difficult) experience.

2) C# is neither a subset nor a superset of C++. C++ will not compile in a C# compiler and C# will not compile in a C++ compiler.

3) Yes, there is a command-line compiler (csc.exe). You pass the path to the file to compile, along with build parameters.

4) Yes, you would have to implement the CLI on the platform of your choice. The CLI is an open specification (meaning, you buy the documents that describe it). It would take a lot of work.