Let's denote the number of prime numbers of the form $3k+1$ which are not greater than $x$ with $\pi _{3k+1}(x)$. Similarly let's denote the number of prime numbers of the form $3k-1$ which are not greater than $x$ with $\pi _{3k-1}(x)$.
1) Are there infinitely integers $x$ such as $\pi _{3k+1}(x)=\pi _{3k-1}(x)$?
2) How often $\pi _{3k+1}(x)=\pi _{3k-1}(x)$ is true? Can we find an asymptotic formula for this?
On
There are arbitrarily large $x$ so that $\pi_{3k+1}(x) > \pi_{3k-1}(x)$, and arbitrarily large $x$ so that $\pi_{3k+1}(x) < \pi_{3k-1}(x)$. And thus there are infinitely many $x$ such that $\pi_{3k+1}(x) = \pi_{3k-1}(x)$. However, $\pi_{3k-1}(x)$ is larger far more often. Asymptotically, the percentage of time that $\pi_{3k-1}(x)$ is larger approaches $100$% as $x \to \infty$. This does not give an asymptotic for the number of times they agree up to some $X$, of course. It's just rather interesting.
This was the subject of a 1914 paper from Littlewood, and I'm distilling information from that paper.
See the paper Chebyshev's Conjecture and the Prime Number Race, by Kevin Ford and Sergei Konyagin. The last paragraph on the first page answers your first question: the quantity $\pi_{3k+1}(x)-\pi_{3k-1}(x)$ changes sign infinitely often, so it must equal zero infinitely often.
I don't have any suggestions for your second question. As Giovanni Resta comments, these problems are difficult.