I don't remember Fire and Ice enough, but that would be my suggestion for the first one as well.
The game with the fat man, well I can only think two games with such a game and character, one is Prehistoric and the other is A prehistoric tale. You are not actually a fat man in either of them but close enough.
The other two games that I can think of that might have been (for a young kid) are Bubble Bobble and Livingstone I Presume. Now, I know that you are not a fat man in any of them but could be seen as one by a kid.
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