Hello Samir
Yes indeed. R2 becomes an ABR between Area 2 and Area 0. It also becomes an ABR between Area 1 and Area 0. It does not become an ABR between Area 1 and Area 2 because by definition, an ABR connects a non-backbone area to the backbone area.
So to answer your second question, it does not propagate Type3 LSAs from Area 1 to Area 2 and visa versa. It only propagates LSAs between the backbone area and a non-backbone area.
It sounds inefficient but that is by definition how OSPF operates. This is also why a virtual link should only be deployed as a temporary solution and not as part of a permanent architecture of a production network.
I hope this has been helpful!
Laz