I don't know if you've made your decision by now, but I would try to avoid flying coach on the Copa flights. I have flown LAX-PTY in coach many times, and PTY-GRU twice with them. They're not terrible, per se - they're pretty organized, actually - but the prospect of two back-to-back 6+ hour flights on a 737 with 31-inch seat pitch and no personal IFE, with only a one hour layover in Panama (an airport with shockingly few food and drink options, though it's getting better) should scare you off. I haven't taken it, but I would do the Korean Air flight in a second.

Copa also flies to Manaus from Panama, but you'll have to spend the night in Panama on the way back to LAX because you can't make the connection.

I would avoid Miami if you can. To this traveler, the Miami airport combines the worst of the USA and Latin America, and the AA flight between LAX and MIA is long and miserable.

I haven't been to either (planning a trip now), but my understanding is that Belem is the more interesting city by far (the Ver-o-peso market being the #1 attraction), but Manaus has better infrastructure for international tourists. I am leaning strongly towards Belem, but that's largely because I've been to the jungle before and I think it's more interesting for cultural reasons.