The Tits classes of simply connected simple real groups

To answer your second question you can use the fact that the Tits algebras of a group are the same as for its anisotropic kernel (J. Tits, Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque, 5.5), and for a quadratic form the Tits algebras are the components of the even Clifford algebra. So for Spin(2m+4q+2,2m-4q-2) the answer is the same as for the compact group Spin(8q+4), and by the Bott periodicity the same as for the compact group Spin(4), that are clearly just two quaternion algebras.

Question 1: I haven't seen an explicit table in the literature of the Tits classes for simple $R$-groups.

That said, such a table can be constructed from tables in the literature. Specifically, the Tits class is determined by the Tits algebras corresponding to the minuscule dominant weights by Proposition 7 in my paper Outer automorphisms of algebraic groups and determining groups by their maximal tori (Michigan Mathematical Journal 61 #2 (2012), 227-237).

These Tits algebras are described for each simple type in general terms for any $k$ in section 27 of The Book of Involutions, and their precise values for $k = \mathbb{R}$ are available in several references, for example Tits's Tabellen zu den einfachen Lie Gruppen und ihren Darstellungen.

Question 2: Victor answered this question in the language of Tits algebras, not the Tits class, so I just translate his answer. Identify the projections on the two components of $H^2(\mathbb{R}, Z_{qs}) \cong \mathbb{Z}/2 \times \mathbb{Z}/2$ with the highest weights of the half-spin representations as in section 4.3 of Tits's paper Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque; this agrees with the identification of $Z_{qs}$ with $\mu_2 \times \mu_2$ that you refer to. For $SO(2m + 4q + 2, 2m - 4q - 2)$, both half-spin representations are quaternionic, so the Tits class is $(1,1)$. For $SO^*(4m)$, one half-spin representation is quaternionic and the other is real, so the Tits class is $(0,1)$ or $(1,0)$.