I cannot at the moment find a way for this one:
Does there exist strictly increasing sequence of primes $q_i$ so that we have $$\sum_{i=1}^{+\infty}\dfrac {1}{q_i+1} \in \mathbb Q$$
I am struggling with something a slightly less general but decided to ask it in this form.
Yes, and in fact you can find such a sequence for any rational $q\in\mathbb{Q}^+$; this is an easy consequence of the fact that $\sum_p \frac1{p+1}\to\infty$. Suppose we want to get a sequence $p_i$ with $\sum_i \frac1{1+p_i}=\frac23$. Then we can proceed in stages: choose $p_1$ as the smallest prime such that $\frac1{1+p_1}\lt \frac23$; then choose $p_2$ as the smallest prime such that $\frac1{1+p_1}+\frac1{1+p_2}\lt\frac23$; etc. You should be able to show (a) that there's always another prime to choose, and (b) that the limit of the sequence is $\frac23$.