Properties of VC dimension

"According to the definition if H doesn't shatter the set of size k, it doesn't shatter any possible set of size k"

-> No, e.g. say your hypothesis space is all straight lines. Then there exist sets of 3 points that can be shattered using this model (any 3 points that are not collinear can be shattered), but some that cannot (3 collinear points).

$H$ = all straight lines:

Keep in mind that the VC dimension of a hypothesis set $H$ is the most points $H$ can shatter.

Your answer to the second question sounds good.

If you have difficulties with understanding the notion of VC dimension, I strongly recommend CalTech's free machine Learning online course by Yaser Abu-Mostafa Learning from Data (the first 7 lectures).