What does "cozy" mean?

What’s does "be up to date" mean? Is it similar with "hand in"?

No, to "be up to date" in this context means that something is new, recent, or current. So in this case here it is referring to the transcripts being new and not old. In other contexts, "up to date" can also mean that something is "modern". Ex: The calendar you have is not up to date. It is from last year. Ex: Your vaccinations are up to date.

What is the role of "happen to" here?

"Happen to" here means "by chance". He is being somewhat sarcastic and saying that he doesn't "by chance" fit the role of a "dumb, insensitive suburban husband", even though that role is not necessarily something that happens "by chance". In other words, he is using "happen to" here to emphasize that he doesn't fit that role.

What does "time on [someone]'s hand" mean? Is it a common expression?

Yes, it's a common expression. It simply means that one has time; they are not busy. Ex: I can help you out with moving this weekend. I have lots of time on my hands recently. Ex: Work has been very busy recently, so I don't have a lot of time on my hands.

Could you explain me the expression "go through" please?

"Go through" here means "undergo" or "experience". The speaker is saying she doesn't want to have a certain experience with Ross again. The present tense of "go through" is "going through", and the past tense is "went through". Ex: I went through a hard time last year. Ex: I'm going through a breakup right now. Ex: I have to go through a medical treatment soon.

What does "wedded to one's gold" mean? Is it an idiom?

Although this sounds like an idiomatic expression, "wedded to one's gold" actually isn't. It's simply a rather descriptive phrase uttered by the speaker to state that the man loves his money - therefore, he's "wedded [married]" to it. "Wedded to X" however, IS an idiomatic expression - it means to be a firm believer of something or to be strongly committed to something. Ex: I tried to get my friend to get an Android phone but he refused. He's wedded to his Apple products. Ex: The politician refused to budge on the issue. He was wedded to the idea of tax exemptions for corporations.