the difference of expression

1>I think that "I can't stand losing you" means 'I don't like to lose you.. Is it right??
If 'can't stand' means 'not like', we use ing form of a verb (not 'to..') after 'can't stand'
so..I wanna know the meaning of 'I can't stand to be lonely'..I wonder if "I can't stand being lonely" is right expression.
I think that it might mean ' I don't like to be lonely'. What do you think??

<2> what does "Let me carry your bag for you" mean?
Does 'carry' mean holding your bag or moving your bag to somewhere?

<3> The customs officer made Sally open her case.. here what does 'case' mean? Does it mean "suitcase' or "box"? If it means 'suitcase', you usually call 'suitcase' just 'case'??

"Can't stand" is an extreme version of "don't like" that really means either "hate" or "can't bear".

Carrying does imply actively moving a bag from point A to point B, not just holding it.

A case can be any kind of case, from suitcase to briefcase. Can't tell from the sentence which is implied. Most people call a suitcase a suitcase, but you might shorten it to "case" if it were already fully established that it was a suitcase. Briefcases are a little more likely to just go by "case" sometimes.

Heath, I am not sure what you mean. You can say "I don't like doing that". In fact, that is more idiomatic than "I don't like to do that".