I happened to run across an answer to 2. (and therefore 1.) just by clicking on one of the related MO links, finding the same question in a comment by Timothy Gowers under <a href="http://mathoverflow.net/questions/16578/can-a-connected-planar-compactum-minus-a-point-be-totally-disconnected">MO16578</a> that was later answered in a comment by Anton Petrunin, and then googling for more information. The key term you want seems to be "<a href="http://en.wikipedia.org/wiki/Pseudo-arc">pseudo-arc</a>", apparently a construction very familiar to the point-set topology community. 

A pseudo-arc is a continuum $X$ (a connected compact metrizable space) with more than one point such that no subcontinuum $A$ (a subspace that is a continuum) is a union of two proper subcontinua of $A$. Remarkably, all pseudo-arcs are homeomorphic to one another!