Let's start with the first question: A connected component is compact in a compact space? I would say yes, because I know that a connected component is closed, and closed implies compact.
I feel like a connected component by arcs is not always compact, but I'm struggling to find a suitable counterexample, something with $sin(\frac{1}{x})$ maybe?
Indeed, the connected components are closed, so compact if the whole space is compact.
Consider $$ X=\{(x,\sin(1/x):x\in(0,1]\}\cup\{(0,0)\} $$ as a subspace of $\mathbb{R}^2$. This set is closed and bounded hence compact.
The subset $X_1=\{(x,\sin(1/x):x\in(0,1]\}$ is arcwise connected, but not closed, because it is dense. The space $X$ is well-known to be connected, but not arcwise connected; the arc-connected component of $X_1$ is $X_1$. See Topologist’s sine curve.