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> 1. Is it true to say that a compact hausdorff space $X$ is path connected if and only if for every continuous function $f:X\to \mathbb{C}$, we have $f(X)\subset \mathbb{C}$ is path connected?
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> 2.For a locally compact Hausdorff space $X$, is it true to say that $X$ is path connected if and only if the Stone Cech compactification $\beta X$is path connected?
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Two questions on path connected spaces
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>


