Edit Here is an easier example. Let $K$ be the middle third Cantor set and
$$B=\{(t,ty): 0\leq t\leq 1, y\in K\}$$
be the cone over $K,$ the Cantor branch and $C=B\cup\varphi(B),$ where $\phi$ is the central symmetry about the midpoint $(1/2,0)$ of the bottom twig $[0,1]\times\{0\}.$ Thus $C$ is obtained by gluing two Cantor branches rotated by $\pi$ relative to each other along their bottom twigs. This space $C$ is compact and pathwise connected, but it is not locally connected at each point. Indeed, a small neighborhood of every point $P$ will contain parts of Cantor twigs not passing through $P.$
Yes. A solenoid is a homogeneous continuum (=compact connected metric space) embeddable in $\mathbb{R}^3$ that is not locally connected at any point, in fact, a small neighborhood of each point looks like the Cantor set crossed with an interval. Its generic projection to $\mathbb{R}^2$ is compact, connected, and not locally connected at each of its points. [Edit I am not longer confident that the last claim is true.]