Does downloading free apps from android app market onto my t-mobile phone charge anything?

Nope. However if an app uses a permission such as send SMS or make phone calls, the app can cause you to be charged money. Typically this would be classified as a malicious app if it did so without your consent and would likely be removed from the market.

T-Mobile generally has unlimited data packages and you would be throttled if you exceed download limits. However, some old data plans may charge overages and even downloading free apps could cause you to be charged overage fees, but this shouldn't be the case with any modern data plans.