If the universe is expanding why doesn't the matter in it expand proportionally making it seem as if the universe is static? Alternatively, as spacetime expands why does it not just slide past matter leaving matter unmoved? What anchors the matter to a particular point in spacetime?
It's tempting to think of spacetime as something like the rubber sheet that is so popular in analogies for spacetime curvature. In that case it's quite reasonable to ask why can't matter slide over the rubber sheet as it expands.
However this is a misleading idea of what spacetime is. Spacetime isn't a physical object, it's a mathematical structure$^1$ that tells us how to calculate the distance between objects, so matter can't slide over spacetime.
If we consider two objects moving apart due to the expansion of space there is no sense in which spacetime is pushing the objects apart. The objects remain at rest and feel no force - it's just that the amount of space in between them is increasing with time.
Re your first question, the reason why matter doesn't expand due to the expansion of spacetime is covered by the question Qmechanic linked: Why does space expansion not expand matter?
$^1$ a combiation of a manifold and a metric