What definition of torus are you using Marco? The usual definition is $T^k = (S^1)^k$ but from the the way you've structured things above it looks like you're using the convention $T^k = S^1 \times S^{k-1}$. Either way, the homotopy-groups of these diffeomorphism groups are generally not trivial as they generally contain plenty of torsion. $Diff(S^1 \times S^{k-1})$ contains $SO_2 \times SO_k \times \Omega SO_k$ for example.
Question 1 is a standard pseudo-isotopy type question. Have you looked up the literature on pseudo-isotopy diffeomorphisms of $S^1 \times S^{n-2}$ ? For example, there's a recent arXiv paper of Crowley and Schick which states that there's elements of $Diff(S^n)$ with large Gromoll Degree, meaning they can be put into positions like in your questions 1 and 2, yet they're non-trivial diffeomorphisms of the sphere.
http://arxiv.org/abs/1204.6474
edit: For some basic results on the homotopy-type of $Diff((S^1)^n)$ see:
Hatcher, A. E. Concordance spaces, higher simple-homotopy theory, and applications. Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 1, pp. 3--21, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978. (Reviewer: Gerald A. Anderson) 57R52
