I would like to understand from a visual/geometrical point of view which kind of transformations you are allowed and not allowed to do by considering these three notions (I would like to be able to recognize them if I see them on a animation for example and to understand their differences).
If we consider the letters $\textsf{O}$ and $\textsf{Q}$ in $\mathbb{R}^2$, then $\textsf{O}$ and $\textsf{Q}$ are homotopy equivalent (we can retract the "bar part" of $\textsf{Q}$ to the intersection point of the "O part" and the "bar part" in order to obtain $\textsf{O}$), but not homeomorphic (if we remove the intersection point of the "O part" and the "bar part" of $\textsf{Q}$, then we have three connected components, whereas if we remove a point of $\textsf{O}$, we only have one connected component), is this correct ? Are there any other transformations that are allowed with homotopy equivalences which are not allowed with homeomorphisms ?
For ambient isotopies, it is even less simple for me... I rode that If we consider the unknot 
and the trefoil knot 
in $\mathbb{R}^3$, then these two are homeomorphic, but not ambient isotopic (within $\mathbb{R}^3$):
For the fact that they are homeomorphic, I know (by definition) that this has to be the case, but I cannot represent myself why is that so. I took a look at this topic: Which two knots are isotopic but not ambient isotopic?, where there is a nice animation which is supposed to explain the situation. My problem is that I cannot understand why it works with the animation (I have the feeling that when $t$ goes to $1$, everything "shrinks" to a point and it makes me think rather about an homotopy equivalence than an homeomorphism...).
For the fact that they are not ambient isotopic, I'm going to be quicker: I don't understand where comes the problem with the previous animation... My only feeling is that, if I imagine a kind of tube around the knot, there is going to be a moment $t'$ where the tube will interesects with itself (a bit like before, when I told that "everything shrinks to a point", but before $t' = 1$).
So, as before, which transformations are allowed with homeomorphisms which are not allowed with ambient isotopies ?
I know that it might be a little bit fuzzy and unprecised, but I clearly have some issues to understand what happens and I wanted to try to explain what I understand about the situation. Thank you for you help.