Please, I'd like to know if I can save the slots in the Live Mode page so that I can have presets with the patches that I need for the songs, or I will have to change them manually everytime I'm gonna play another song.

As far as I can tell, it seems like you should be able to save a Live Mode setup as part of a Multi. Then you'd save a different Multi for each song, and each Multi would include all the patches set up in Live Mode for a particular song. I don't have Omnisphere here to check that, but I'll attach one of our Omnisphere courses to this thread -- there are several videos that cover the organization of Patches, Parts, and Multis (videos 4,5,6), and creating a Multi (video 33), as well as one on Live Mode (video 35).