Determining if (infinite) binary language DFAs contain at least 1 prime?
It's a standard intro theory exercise that for any $d\ge 0$ there's a FA that accepts all and only those strings in $\{0, 1\}^*$ that are the binary representations of integer multiples of $d$. Thus, the answer to your second question is "yes".

For your third question, the answer is "no". Consider the regular language denoted by $1(10)^*0$. This language consists of those binary representations of
$$
\frac{10\cdot 4^n-4}{3}
$$
and clearly this sequence is too sparse to contain an arithmetic subsequence of any length greater than 2.