Hey, I know Manaus and Presidente Figueiredo well since I lived there for several months.
First off, forget the idea of a "trek" in the Amazon...
The best option is to go straight to Pres. Figueiredo and stay there.
The waterfalls are easily accessible, and you can sleep at many sites, usually in a hammock.
As for animals, well, you gotta get up early :) and paradoxically, the place where you’ll see the most animals like monkeys, parrots, toucans... is at Urubui Park, the most visited spot in Pres. Figueiredo.
For waterfalls, the most spectacular ones are Mutum, Iraceima, Santuário, Orquídeas, and Natal.
There are a lot of them... one week is really short for this kind of place.
As for Manaus, apart from MUSA, the theater (there are usually small free performances to see in the theater on Fridays around 7 PM), and Ponta Negra, you can stop by the CIGS zoo on the way to Ponta Negra. (MUSA and Ponta Negra are pretty far from the city center.)
Manaus is a good city for shopping.
You can access several beaches on the Rio Negro from Ponta Negra.
For hotels, I always stay at Hotel 10 de Julho, which is right next to the opera house and is basic but cheap.
If you want more info on hotels, I wrote a post about it here:
https://voyageforum.com/forum/retour-hotels-manaus-d8308177/
For getting to Figueiredo, you can take a bus or a shared taxi. In either case, it’ll take about 1.5 hours.
Pousada da Wal in Figueiredo is the only hotel I’ve tried, and it’s great.
If you need more info, don’t hesitate to ask.