There are plenty of hotels in Sayaxche, including the following from Tripadvisor:
https://www.tripadvisor.com/Hotel_Review-g424968-d1159249-Reviews-Hotel_Ecologico_Posada_Caribe-Sayaxche_Peten_Department.html
https://www.tripadvisor.com/ShowTopic-g424968-i29197-k6777728-Hotel_del_Rio_Sayaxche-Sayaxche_Peten_Department.html
I don't have any personal experience with them. I wouldn't worry too much about booking ahead, even if you can't find something groovy and atmospheric, Sayaxche is a fairly busy town and I'm sure you could turn up a decent place to stay on short notice.
Uaxactún/Tikal
Have you done a trip like this and do you have any recommendations on who should I ask to organize this kind of trip? My idea is visiting Uaxactún in the morning then spend the night in Tikal.
Is it possible to sleep in Tikal on a budget? Like renting a tent or a hammock?
I've read that some of you did the same but every hotel I've found offer only rooms...
The only public transportation serving Uaxactun would not make such a trip possible, so you would need to arrange it with a tour agency or (better) a private driver. I have had excellent luck arranging for trips with a private driver at the Cafe Yaxha--the owner is a well-connected archaeology buff. The price for a full day with a private driver will be more reasonable if you can find a group to make the trip with you, of course.
If you have plenty of time, you could do this trip by bus, but it would be better to visit Tikal first. The bus goes from Flores to Uaxactun in the afternoon, stopping at Tikal around 3 or 3:30. Tickets bought then are good for the next day, so you can poke around for the afternoon, sleep in the park (see below), visit more extensively the next day, and then catch the bus up to Uaxactun in the late afternoon. There are basic places to stay in Uaxactun. You could spend the night there, tour the ruins all day, spend that night as well, and then catch the bus back to Flores at 6am. Note that doing it this way would require dedicating 3 nights.
Renting hammocks or tents at Tikal is easy. There is a government-run campground in the same complex as the hotels, which is very pleasant. You can rent a hammock with mosquito net (they'll set it up for you) for $10 US. I highly recommend it.
Yaxhá/Topoxté
The same question. Is it possible to rent a tent or hammock?
Yes, there is a campground at Yaxha.
How can we reach Topoxté? Do we have to make a reservation for that or can we arrange it in place?
I'm sure you can arrange it with somebody at El Sombrero, the hotel on the lake. I would not worry about setting it up ahead of time unless you are visiting the site with a guided group, which would restrict your ability to make last-minute decisions.
Is it an option to travel to Yaxhá FROM Tikalor to Tikal FROM Yaxhá or in every case I have to go back to Flores and then change my direction? If there is a way to do a Yaxhá - Tikal/Tikal Yaxhá trip could you give me some more information on that?
Gotta go back to Flores, or at least to the crossroads at Ixlu, where you can hop on a colectivo headed to Yaxha. Note that public transportation does not actually go all the way to the ruins at Yaxha, so you'll need either a private vehicle, a tour group, or some luck hitching a ride.