You need to express the past event:
It's been a month since the deadline (...)
and the future event:
... and it's only one more month until the program starts.
Simple present will do, as you are talking about a scheduled event (See CopperKettle's answer here).
Don't be tempted to mash them together!
For improved understanding, the "only one more" helps to clarify that you are talking about two different months. The bit in italics is where you would express your "attitude" or "view" towards the upcoming time frame - "still another" / "even another" / "one more"/ "just one" / ... - there are lots of options.
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.
Best Answer
This is probably testing if you know that the simple present can be used to refer to future events, when they happen certainly, due to a calendar or schedule. Birthdays are an example.
You can also say "The bus leaves at 9pm" or "I graduate next year". These events are seen as being inevitable due to the bus timetable, or the University schedule.
So that is probably what this question is about. But all three are syntactically correct. All three express the same general meaning, in different ways. None of the options is clearly incorrect.
"Will" expresses future events, it is a fairly neutral future tense. "Going to" tends to be used for future events that occur as a result of present choices and conditions. "Going to" would be the least appropriate, as the birthday won't occur as a result of the choices we make today. It will happen anyway.