This use of “nip” is more common in British English than American English, so I will let a BrE speaker address that part of your question. Other options: I’m going to go down and get Tom. I’m going to dash down and let Tom in. I’m going to pop out for a minute to let Tom in.

Just to reinforce SuKi’s last point, you would never hear this in US English. People would have no idea what you mean. We’d: I’ll run down and get Tom/let Tom in. Or maybe, but less common (but more similar to “nip”) I’ll pop down and get Tom/let Tom in.

"I think I might have a nip of brandy." Which basically means, "I'd like to try a small amount of brandy" or "I think I'll have a small amount of brandy." I'm not sure how commonly "nip" is used this way in America, anymore, but I hear it (and use it) like this, sometimes.

I'm glad to see that you're getting to grips with nips! As with most verbs of movement (come, go, run and so on), you can combine 'nip' with a range of adverbs to indicate both the manner and direction of movement. In the scenario you describe, I'd say 'I'll nip down and get Tom' or 'I'll nip down and let Tom in' or (as you suggest) 'I'll nip down for Tom' is also possible. It's also possible to say 'nip out' in those sentences (meaning leaving the party/leaving the flat), but 'nip down' (meaning going downstairs) describes better what you're doing. You could use 'to' as an infinitive of purpose, but 'and' sounds more natural in these sentences. Note the you wouldn't say 'bring Tom over' in this scenario. 'Bring over' would suggest that you were bringing him, say, from one end of the town to another, as in "Why don't you bring your friend over for dinner some time?'. 'Over' indicates some distance between the two places in question, so we wouldn't use 'over' if he were simply standing at the front door of your building. You could say 'bring him up', but I think we'd probably just use 'get'. NB As I pointed out before, this usage is British English. An American English speaker would not use 'nip' in this way.

"nip it in the bud", "nip something in the bud" = stop something before it gets big, widespread, common, ... John's answer is also good. "pop" and "nip" are used in very similar ways. BTW: Tom's car is out of action. I'll bring him over when I come.