Question 1 Answer: Lets assume that your included diagram is an accurate example of how a wormhole would function. What you must realize when studying this model, is that it does NOT represent a wormhole in three dimensional space, but two dimensional space. So when you imagine a ship flying through this wormhole, you have to imagine a two dimensional ship, completely flat like paper, that can ONLY fly WITHIN the yellow surface. The yellow surface IS its universe, and it cannot leave.
In simpler terms, get some paper and draw a ship larger than the hole, cut it out, and slide it along the yellow surface. All parts of the drawn ship must always stay in contact with the yellow surface. Imagine trying to slide this paper ship through the hole. If the ship is slightly larger than the hole, and the paper is flexible enough, the nose of the ship will slide through the hole, but the top of the ship will wrap around to hit the bottom of the ship.
The nose (front) of the ship would never touch the tail (back), it would curl sort of like a hot dog, and the top and bottom would but up against each other. If the ship was going very slow, it would simply not be able to pass through, but if it was going very fast, it could catastrophically damage itself.
Now if we forward this example onto three dimensional space, things get kind of cool. Imagine you have a football-shaped ship, that is slightly larger than a wormhole. As it attempts to pass through, the tip would go through, but it would get stuck, and this is the cool part. Not only would the top of the football butt up against the bottom, but simultaneously the left would hit the right, the top-right would hit the bottom-left, the bottom-right would hit the top left. The entire ring of the football would touch itself on all sides simultaneously! (Assuming the cut-out shape of the football was perfectly circular, and the 'shape' of the wormhole was perfectly 'circular')
Now I started this answer extremely intrigued by your second and third questions, but not having any answer in mind. But as I typed that first answer, I think I figured it out! If anyone knows more about questions 2 and 3, or has any corrections for me, I would LOVE to know!
Tentative Question 2 Answer: You are almost correct in your thought process of space-time curvature and how it affects materials. If a curvature in 2D space bends the 2D object through the third spacial dimension (as in your model), then a curvature in 3D space will bend the object through the fourth spacial dimension. This is fascinating, since we cannot observe the fourth spacial dimension, and we know that space-time curvature manifests as gravitational effects on objects, then it seems to suggest that bending a 3D object through a fourth spacial dimension would be equivalent to subjecting it to an acceleration. Thus our observation of gravitational effects is simply how we interpret objects bending through 4D space.
So to answer your question, the material might very well be cracked or pulverized by too much 4D 'bending', just as a very large variation in acceleration (as near a black hole) could rip an object apart!
Tentative Question 3 Answer: Given the conclusions set forth in the previous answer, I believe this 'resistance' to 4D spacial bending is what we observe and commonly know as 'inertia', which is defined as a resistance to a change in velocity. Which can be called a resistance to the effects of acceleration, or, a resistance to gravitational effects, or, a resistance to 4D 'bending'.
Since you asked this on a world building stack exchange, I am going to assume you are writing a book or something of that nature. Might I suggest that you subtly allude to that fact that whatever mechanism creates wormholes in your world 'shapes' the wormholes so that there exists a path through it that avoids any sharp curvatures, thereby bypassing the problem of extreme gravitational effects ripping objects apart near sharp space-time curvatures.