Yes, that is correct. It is a free abelian group on two generators.
Yes.
Since is simply connected, .
The free product of a free group and a free group is a free group having a presentation , where = 1 or -1 in the first and = 1 or -1 in the second .
For instance,
(followed by a member in the first group) .... ,
reduces to
,
where denotes the element x in the i-th , i=1 or 2, and the group operation between elements in the same group is an addition.
As you know, and are two different things. The former is non-abelian because a free group on a nonempty set S is abelian iff S has exactly one element. The latter group is abelian, which is isomorphic to the fundamental group of a torus having a presentation .