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.
I would say the best fit is are you related?.
Related means 'belonging to the same family'. This can be rather broad. I am obviously related to my brother, but I am also related to my great great uncle's son on my mother's side.
Best Answer
"Politely" has various levels so I'll try to address several:
This is probably very British. It's very polite. Notice we haven't actually asked for the room to be cleaned, we have asked if it's a problem. If the person you're asking is British, this is the most polite method. You can even experiment with words like "awfully" (Would you mind awfully...?), or by adding other conditionals (...if you have time).
Are my firm favourites for sounding a) polite and b) not too polite as to sound foolish.
Lastly, I offer a more familiar (though polite) option (assume you have known the person a while but in a respectful/businesslike way):
Hoover = British English
Vacuum = American English