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 MO16578 that was later answered in a comment by Anton Petrunin, and then googling for more information. The key term you want seems to be "pseudo-arc", 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!

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 MO16578 that was later answered in a comment by Anton Petrunin, and then googling for more information. The key term you want seems to be "pseudo-arc", 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!

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 MO16578 that was later answered in a comment by Anton Petrunin, and then googling for more information. The key term you want seems to be "pseudo-arc", apparently a construction very familiar to the point-set topology community.

A pseudo-arc is a continuum $X$ (a connected compact metrizable space) 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!

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 MO16578 that was later answered in a comment by Anton Petrunin, and then googling for more information. The key term you want seems to be "pseudo-arc", 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!