Let's break this question into two pieces.
What's Redundant
The exercise instructs you to cross out redundant words in the sentence. Redundant means not needed or superfluous.
I'd argue that the entire beginning of the sentence is redundant because someone reading the sentence already knows that you're writing to tell them something. In American English at least, you can also omit the and between come and stay as well (ht: Robusto).
I’m writing to tell you that I can’t come and stay with you the next weekend.
Next Weekend vs. The Next Weekend
I don't think there's enough context to know whether the sentence should be "next weekend" or "the next weekend," since they mean different things. I suspect it's the former, but it's just a guess, and it's worth a short discussion about the difference.
English has a slight, unfortunate ambiguity in how the near future is described:
- "this weekend" = current weekend or first upcoming weekend
- "next weekend" = first or second upcoming weekend
- "the next weekend" = first weekend after some previously referenced date
Example conversation:
A: Do you want to meet next weekend?
B: Do you mean this weekend or the following weekend?
A: The following weekend.
B: Sorry, I can't make it. How about the first weekend next month?
A: Can't do it, but I can meet the next weekend.
B: That works. See you then.
So, getting back to your question, it's possible that "the next weekend" makes sense, if they're talking about the weekend following some previously discussed date. It seems unlikely, though, and I tend to agree that it should be "next weekend."
The same rule applies to your other examples. The apology happens "the next day", i.e. the day after the argument. The doctor calls next week, i.e. the first upcoming week.
You're right, both possibilities are vague. I would argue that they are equally vague. There are no significant differences between using "next" and "in the next" if you want the time frame to be unclear.
If, however, you are concerned because you want the time frame to be more defined, it might be helpful for you to consider ways to add specificity. For example,
I will meet you one week from today.
For more great examples, see this post: "In the next two weeks" vs. "next two weeks"
Best Answer
The definite article is used when specifying a specific instance of a set, where it is expected that both the speaker and the listener know which instance is being referred to, and the instance is not a proper name.
By convention, immediately approaching days of the week are treated, grammatically, as if they were proper names, and thus the definite article is not used.
This is acceptable, assuming that we are talking about the immediate next Monday that appears on the calendar.
This is acceptable, as we are no longer talking about a single Monday, but about all Mondays.
This is not acceptable. We need the definite article, because we are speaking of a specific Monday, that is not a proper name (that is, is not the immediate next one).
This is now correct. If today is Sunday, March 1st, this refers to Monday, March 9th. If today is Tuesday, March 3rd, it refers to Monday, March 16th.
This is also correct, but subtly different in meaning. If today is Sunday, March 1st, this refers to Monday, March 16th. If today is Tuesday, March 3rd, it refers to the same day, Monday, March 16th.
This is correct, for the same reasons as for days.
This is incorrect, but you might hear it from time to time in spoken English anyway. Spoken English often includes grammatical mistakes. Written English is supposed to be better, but mistakes happen there as well. This particular mistake can be parsed as correct grammar if you treat "week before last" as a noun phrase, but this is definitely non-standard and non-idiomatic.