Im confused about Sinatra (the ruby framework).
Is it a lightweight Rails replacement or you can have them running side by side?
Can you do a web application (as in Rails)? For example a twitter clone?
Sinatra is not Rails. It is a micro-framework used for simple websites where you may need to just define a few actions. You can make a Sinatra application as complex as you want to, but you'll hit a point where you code has become a crazy mess sooner than with Rails.
While not 100% accurate, Sinatra fits mostly into the Page Controller architectural pattern, and Rails is a clear MVC implementation.
To answer your questions specifically: