It is known that there exists complete intersection of given type $T=(d_1,\ldots,d_c;n)$ with infinite automorphism group if and only if the type satisfies one of the following:
$$T \in \{(2;n), (3;1), (2,2;1), (4;2), (2,3;2), (2,2,2;2) \}.$$
I.e. quadrics, curves of genus $1$ and K3 surfaces. In all other cases, the automorphism group is always finite.

See for example Theorem 3.1 of:

Olivier Benoist - Séparation et propriété de Deligne-Mumford des champs de modules d'intersections complètes lisses.

As for your other question; yes there are formulae for $\mathrm{H}^1(X, T_X)$ in terms of the degrees of its defining equations. They get quite messy though. I'm not sure where to find these; perhaps in SGAVII or Hirzebruch's book "Topological methods in algebraic geometry".

In the case of intersections of two quadrics however it is quite easy to work out. First you can use the Euler sequence and standard cohomology computations as in this question Deformations of smooth projective hypersurfaces and the Jacobian ring , but there is also a nice heuristic argument. Namely, a complete intersections of two quadrics of dimension $n$ is determined by the discriminant of its associated pencil of quadrics, which is a closed subscheme of $\mathbb{P}^1$ of degree $n+3$. These have $n+3-3=n$ moduli. Hence we find that
$$\mathrm{H}^1(X, T_X) = n.$$

Many useful facts about intersections of two quadrics can be found in Miles Reid's PhD thesis: http://homepages.warwick.ac.uk/~masda/3folds/qu.pdf