The first one is good. The second one is basically incorrect or means something different. I feel good doing sports = I like doing sports; I feel good while doing sports. I feel good to do sports = I feel phsyically capabable of doing sports (usually with 'good enough').

‘I feel great when I do sports’, not ‘to do’. The second example doesn’t make sense grammatically.

#1 is correct and easy to analyze. "To do sports in my free time" is a noun clause that acts as a placeholder for "it". In fact, the sentence can be rewritten like this: " To do sports in my free time feels great ." #2 is impossible to analyze and is therefore a bad sentence. The phrase "to do sports in my free time" plays no useful role. It does not modify "I' or "feel" or great" so it serves no clear purpose. Here are some ways to fix the sentence: A) To modify "feel", say "when I do sports in my free time". That gives you an adverbial clause answering a "When?" question to state "when you feel". B) To modify "I", turn it into an adjective phrase and put it right next to"I": "Doing sports in my free time, I feel great." C) To modify "great" is a bit more difficult. Turn it into adverbial phrase placed next to the adjective "great": "in my free time, I feel great athletically with sports ."

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