What does "something" mean here? Is "trust someone with something" an idiom?

Does he mean that he likes to sit in a tub with strangers?

No, he means this in a sarcastic way. The way he expresses the experience ("moist" and "with strangers") shows that he finds this action undesirable. You can also pick up clues from his facial expression and flat tone of voice, which is devoid of excitement. "Nothing like X" can be used in a positive or negative way. Ex: Nothing like being late on your first day of work. Ex: Nothing like a nice cold ice cream on a hot summer day.

Is "gosh" same as "god?"

Yeah, that's right! Although it's considered a much milder version of "god." Ex: Oh, gosh. I left my umbrella at home. Ex: Oh my gosh. Please, be quiet. = Oh my god. Please, be quiet.

I'm not sure why "were to" was inserted in this sentence

"Were/was to" is often used to introduce a hypothetical or conditional situation, usually one that is unlikely to happen. Note that this must always be used with "if." Here, the speaker is introducing an if clause, about what she was to told do if something happened. Ex: If I ever were to win the lottery, I would start a charity. Ex: If he ever were to apologize, it'd be the shock of my life.

What does "something" mean here? Is "trust someone with something" an idiom?

Here, "something" refers to information- more specifically, Paul's thoughts. To "trust someone with something" is a phrasal verb! It means that they will be careful and considerate with what you "trust" them with. You consider that person reliable to share information with. Ex: Sarah trusted you with her secret, and you told everyone! You're not a good friend. Ex: Can I trust you to tell me the truth?

Is the speaker really saying "let's see what"? Please check the script.

Yes. Aladdin did say "Let's see what we can do about your situation here."