Learn English – Formal way of asking “When are you in Berlin?”

Q1. I believe both phrases are proper (US) English, though for quite different situations.

Joining someone for a (alcoholic) drink implies a normal, casual social occasion. "Hey, we're going to stop by Chili's on the way home, if you want to join us for a drink." (Or "join us for drinks" implying a period of time.)

Joining someone in a drink would imply a much deeper meaning to your participation. It would emphasize that particular drink and some kind of bonding or ceremonial event. For example, a good friend just got fired and you meet him at a bar and he has obviously been drinking and hands you a beer. Or your boss is very pleased at a huge deal that you just sealed, so he hands you a celebratory glass of whisky and the two of you drink in celebration.

Q2. Unless you're making a toast or something formal, you'd ask someone to join you for a drink (or for drinks). The exact wording of a formal invitation would depend on how formal you mean (e.g. do you mean printed invitation cards?).

Q3. This is more of a cultural question in many of its details, but I would make it more polite not by being formal in my speech but by expressing my gratitude and best wishes for the evening. If I were the sole host and was not married, I would say "I", otherwise I would say "we" (i.e. either my wife and I are the hosts, or I am speaking on behalf of my company or association). For example (making a lot of assumptions about the physical situation, place and time):

"Excuse me lades and gentlemen... Thank you for being here this evening as our guests. We're so happy to see each one of you, and hope that you're already enjoying the evening and each other's company. The food is ready, so if you could please make your way to the dining room, where we'll continue our conversations over some incredible food... Thank you."

I'm glossing over many social/cultural issues here, of course.

In U.S. English, the simplest way would be to ask

What time does she get off work?

A shade more formal would be

What time does she usually leave work?

or

What time does she usually leave for the day?

In the latter, the work context would be understood. Don't use it if there could be some confusion.

Here's one way to ask involving the word stay:

How late does she usually stay [at work]?

Be careful, though, because any of these might be construed by the listener as evidence that you may be nosy at least, and possibly a stalker. So unless you're checking up on this woman as a job applicant, the prudent course would be to ask her yourself.

addendum for OP

You asked where you should place the until in your constructions. Presumably you're wondering whether you should avoid putting it at the end of the sentence. The answer is: don't avoid that. While some pedants will wince at this, they really don't have a leg to stand on. See this accepted answer on ELU to the same question. (TL;DR: This is a rule made up hundreds of years ago by fussbudgets who determined, against all usage evidence to the contrary, that English should behave more like Latin.)

Therefore you should feel no qualms about asking

What time does she work till?

or

What time does she work until?

But since English is pretty flexible, you could also easily say

Until what time does she work?

But for my money, that sounds a little more formal, even stilted.

Note that till and until are pretty much interchangeable here. See this related discussion on till and until on ELU.