How to prove the following two congruences?

Question1: Let $p\equiv 1 \pmod 3$ be a prime, then $$\sum_{k=0\atop k\neq(p-1)/3}^{(p-1)/2}\frac{\binom{2k}k}{3k+1}\equiv 0 \pmod p.$$

Question2: For any odd prime $p>3$, we have

$$\sum_{k=1}^{(p-1)/2}\frac{1}{k\binom{2k}k}\sum_{j=1}^k\frac{\binom{2j}j}{j}\equiv \frac{1}3B_{p-2}\left(\frac{1}3\right) \pmod p,$$
where $B_n(x)$ are the Bernoulli polynomials defined by
$$\sum_{n=0}^{\infty}B_n(x)\frac{t^n}{n!}=\frac{te^{xt}}{e^t-1}.$$