Saying 'You can stay until next year' in December would be either a slightly rude way of saying, 'leave in a month', or a reasonable way of saying, 'leave sometime during the next calendar year.'
'You can stay until a week later' Not a phrase I'd use. It doesn't make sense by itself. 'You can stay for a week' means you can stay for approximately 7 days.
'He did not talk to me until a year later' indicates that no conversation took place for approximately 365 days since the time in question. Your example doesn't include the context, so it's unclear when the follow up conversation took place. It would have to include something to set the starting point of this 365 day gap in conversation. For example, "In May 2008, I called him a jerk. He did not talk to me until a year later." would suggest they spoke again in May 2009.
'He did not talk to me until the next year.' As in the first example, 'the next year' refers to the next calendar year. As in the previous example, it depends on when they say it.
If they say, "In May 2008, I called him a jerk. He did not talk to me until the next year." That suggests they may have talked as early as January 2009. Most likely it means sometime between January and April 2009. Anything longer than that, but still in 2009, would be 'for a year' or 'for more than a year'.
Typically, you wouldn't use this phrase for time periods of less than a month or two. Saying this in December to mean 'a few days' would be pedantic. That said, it's common to hear people jokingly say, 'see you next year' in late December if they don't expect to see the other person before the New Year. Personally, I'd avoid that usage.