I feel like I'm reading the rules correctly, but just double-checking with the community.

: an investigator can commit cards in hand to skill tests during any phase of the game, right? For example, if an investigator has drawn a card during the method phase and is resolving a test, he/she can commit a card to aid in a test?

Assuming the card can be legally committed to the test, yes, you can commit any number of cards to your own tests, and one to another investigator at your location.

Skill tests are skill tests regardless of when/how they occur. So if you drew a card in the clean up phase and then get an encounter card in the mythos phase that requires a skill check, you can commit the card you just drew, or any other card, if they are applicable. Same thing if something happens at any other time in the game.