1. Yes, 'li' is omitted exactly when preceded by 'mi' or 'sina' standing alone (and 'o', but that is later). If you add something to 'mi', either before or after, the omission does not occur (skipping technical explanation).
2.'tawa' is a preposition, the characteristic of which is that the "object" follows immediately after, without 'e'. This is true even when the preposition is doing a verb's job, as here (at least partly). cf. 'mi lon ma tomo Sanluwi" "I am in St. Louis.", etc.
3. Same answer.
kama pona tawa toki pona! (a different construction, but still a prep with object immediately after).