For problem 2 the answer is false for most spaces that one wants to consider. If $X$ is a path-connected paracompact space of non-measurable cardinality, then $X$ is a path component of the Stone-Cech compactification $\beta X$. See, for example, Theorem 3 in the paper On fundamental groups of compact Hausdorff spaces by James Keesling and Yuli Rudyak.

http://www.ams.org/journals/proc/2007-135-08/S0002-9939-07-08696-0/S0002-9939-07-08696-0.pdf

For problem 2 the answer is false for most spaces that one wants to consider. If $X$ is a path-connected paracompact space of non-measurable cardinality, then $X$ is a path component of the Stone-Cech compactification $\beta X$. See, for example, Theorem 3 in the paper On fundamental groups of compact Hausdorff spaces by James Keesling and Yuli Rudyak.

https://www.ams.org/journals/proc/2007-135-08/S0002-9939-07-08696-0/S0002-9939-07-08696-0.pdf