1. In "my homework finished" the "finished" is passive. You did the action.
In "my homework having finished" the homework is the actor. That doesn't make any sense. Your homework did not do anything, so it did not finish.
2. The idea is that it is a nice day, so we should enjoy the outside. You don't want to "waste" the day by staying inside. If you say "it's such a nice day to waste," that means you want to waste the day. Which doesn't make any sense.
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