- Winning is not everything. (does "win" work?)
Yes, but as a verb it would need to be in the to-infinitive form: "To win is not everything". It sounds very formal, but sayings like this often are - for example "to know is to understand". As a noun it would be "a win is not everything".
- Please return it after use. (does "using" work?)
"Using" could work, but only as a verb. As this is an instruction or command - the kind you might see on a sign or notice - the strict rules of grammar do not always apply. As a noun, "use" is countable, so if you wanted it to be grammatical rather than instructional you might want to say "after each use".
- Walking is a good form of exercise for both young and old. (does "walk" work?)
"Walk" as a verb doesn't sound right. It would have to be "To walk is good exercise", which may be grammatically correct as in your first question, but it isn't at all idiomatic, I can't imagine anyone saying it. As a noun, it would be "a walk is a good form of excercise.