I have a question too. If we take this molecule (https://www.wolframalpha.com/input?i=smiles+identifier+C1C%28%5BC%2B%5D%28C%29%28C%29%29C1), will it do ring expansion? Or will sigma-resonance stabilize it enough?

In real life, both can occur.

In an exam-type situation, if you are faced with expanding a 4-membered ring to a 5-membered ring *or* doing a methyl shift to give a tertiary carbocation, you should expand out the 4-membered ring due to relief of ring strain.

There is very little relief of ring strain on going from 5- to 6-, and for that reason the two possibilities are a lot less clear cut in that case.

Almost certainly the hydride would migrate preferentially. It’s just easier from a principle of least motion standpoint.

Do you mean ring contraction? Unlikely to go from five to four since four membered rings have a lot of strain.

If there is competing alkyl vs. hydride shifts that give equivalently stable carbocation, hydride shift will generally be favored since less molecular motion is involved.

Generally, if one is going from a more strained ring to a less strained ring, the reaction should be exothermic since ring strain is being released.