Welcome to the board, Den.
Cruising through the board you will likely find answers to many of your questions. Try using the "search" button to narrow some of threads you might want to look at if you have specific questions that might be answered by a single thread.
For a general question such as what are the advantages of the rotary engine, you will likely find the answers scattered throughout various threads, but try the search anyway using key words you might think would narrow the field to get the answer you want.
Having said all that, and being a rotary enthusiast, I can give you a couple of advantages, and maybe some of the other guys will chip in... 'cause many of us love our rotaries!
From a driver's standpoint, most of the advantages involve the incredible smoothness of the engine, its willingness to rev, and its ability to pull with seemingly ever increasing pull all the way to and past redline. You know how when you rev a piston engine towards the redline and the "pull" you feel (particularly on lower powered engines) seems to fall off just a bit before the redline (that's how 'intuitively' you know when to shift)? The rotary doesn't do that. The pull seems to increase powerfully the more the engine is revved, which is why the engine has a redline buzzer, and why it is recommended to keep an eye on the tach...
Also, the engine is so smooth all the way up that there are few sensory indications (save sound) as to how high you are revving the thing...
Other advantages include small size and low weight. This allowed the engine to be placed pretty far back (behind the front axel, in fact) in the engine compartment, giving the car better balance and lower weight.
The rotary needs to be driven to be appreciated, and there are disadvantages as well. I have always driven high revving engines, and enjoy that type of driving. Hopefully you will find the answers you are looking for on this board...
Beach Resort Researcher