This is a partial answer to Q2 that has nothing to do with interpolation. It concerns the case of finitely supported measures.
Here I am not trying to give general criteria for $\|T\|<1$, as this is a hard question, only to relate this "spectral gap" property for different $p$'s.

**Remark:** Currently I am still unsure about the general case (measures which are not finitely supported). This is incompatible with a comment I made to the original post, which was too optimistic. However using the method Mikael indicates in a comment to this post one can say something, namely that the property of having $\|T^n\|<1$ for some $n$ holds for every $p\in (1,2)$.

Let $G$ be a compact group, $\mu$ a finitely supported probability measure on $G$. Let $T_p:L^0_p(G)\to L^0_p(G)$ be the operator given by left convolution with $\mu$.

**Claim:** Either for every $1<p<\infty$, $\|T_p\|=1$ or for every $1<p<\infty$, $\|T_p\|<1$.

Let $\Gamma<G$ be the (countable) group generated by the support of $\mu$.
We assume as we may that $\Gamma$ is dense in $G$, otherwise $\|T_p\|=1$ for all $p$. It follows that there are no $\Gamma$-invariant vectors. By strict convexity of the norm we conclude that there are no $T_p$-invariant vectors. Moreover, by the uniform convexity of the norm it is not hard to see the equivalence (for a given $p$): $T_p$ has almost invariant vectors iff $\Gamma$ has almost invariant vectors.
The former is equivalent to $\|T_p\|=1$. However the latter is independent of $p\in (1,\infty)$.
Indeed,
for every $p,q$ we can define the (non-linear) Mazur map $L_p\to L_q$, $f\mapsto \text{sgn}(f)|f|^{p/q}$ which has two nice properties: it is uniformly continuous on the sphere and it commutes with isometries, eg with $\Gamma$ (this is for $p\neq 2$, by Banach-Lamperti).
For more details see section 4 in http://arxiv.org/pdf/math/0506361v2.pdf.

It follows that the statement $\|T_p\|=1$ is independent of $p$ which proves the claim.

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