I have two questions: Let $ M$ be a compact connected manifold with boundary.

1, If the boundary $ \partial\tilde{M} $ of universal covering $ \tilde{M}$ is connected, is $ \partial M$ connected? How about converse direction, if not, any counterexamples?

2, Does connectedness of boundary $ \partial M$ imply $ \pi_1(M,\partial M)=0$ , if not, any counterexamples?

If $ M$ is not necessarily compact, will it be different?

Thanks for your help.