Both expressions don't sound English. I would go with:
The project will be completed within a few days.
So without the next. Since you say will be, it implies the action will take place in the future.
Alternatively you could say:
The project will be completed within the next few days.
So with a the.
Next week, with no article, always refers to the week after the week of "utterance time"—the time when the words are spoken or written. For instance:
If I say to you now, on Thursday July 29, "John will answer next week" . . . what I mean by next week is the week after this week, in which my utterance time falls: August 3-9.
If I say to you now, on Thursday July 29, "John said on Thursday July 2 'I will answer next week'" . . . Next week is part of John's quoted utterance, so what he meant was the week after his utterance time: July 6-12.
But if I say to you now, on Thursday July 29, "John told me on Thursday July 2 that he would answer next week" . . . I am not reporting John's actual utterance, but only the substance of his utterance, so next week is part of my utterance, and what I mean is again the week after my utterance time: July 6-12.
The following week or the next week† ordinarily refers to the week after "reference time"—the time which the speaker or writer is talking about.
If I say to you now, on Thursday July 29, "John told me on Thursday July 2 that he would answer the following week" . . . I am now reporting the substance of #2, a conversation on July 2, which is my reference time, and the following week refers to the week after that: July 6-12
If I say to you now, on Thursday July 29, "John always puts things off. If you tell him now that you will ask again on September 1, when September 1 rolls around he will still tell that he will answer the next week" ... Now I am reporting the substance of a conversation in the future, on September 1, which is my reference time, and the next week refers to the week after that: September 7-13.
With a report of a present utterance, you may revert to the version without the article, because in that case reference time and utterance time are identical.
John says that he will answer next week.
With 'generic' present-tense utterances, which are not tied to a specific reference time but to all reference times, you may use either version
It doesn't matter when you ask, John always tells you he will answer { next week / the following week }.
The issue you raise about "the following week" involving a two-week span rests on a misunderstanding. If I say
Next week I will go to London, and the following week I will go to Paris.
I start by talking about one week from now, which becomes my reference time, and the next clause uses the following week because I am now talking about the week after my reference time. Here we are indeed talking about a time two weeks after my utterance time, now.
But if I say
Two weeks ago I went to New York, and the following week I went to Chicago.
My reference time is two weeks ago, and the following week is one week after that ... but it is only one week from my utterance time--last week, in fact.
Next week = the week after utterance time.
The next week or the following week = the week after reference time.
† Some people will tell you that you should always say the following week, because the next week means "during the week starting with the utterance". Unhappily, the only people who actually distinguish the two that way are the people who believe it's a rule, and there are lots of other people who say that it's not a rule and never has been. In practice, then you can't count on the rule, since both version are equally ambiguous; so if you have any doubt, and if it matters, ask what exactly is meant. And there's no reason why you should follow the "rule" either; if you prefer next week, go ahead and use it, and if anyone seeks to "correct" you, tell them that your usage is governed by prosodic euphony rather than the petty pomposities of a preposterous and presumptuous prescriptivism.
Best Answer
My native speaker instinct agrees with your interpretation and not that of your friend. For me the difference between "within 100-200 days" and "within 200 days" is an expectation that the timeframe could be significantly shorter than 200 days but an acknowledgement that 100 days is a reasonable estimate of the likely timeframe. This phrase might be used to ask for something within 100 days but also add an upper bound of 200 days if something gets delayed. If a boss gave a project to someone and asked for it to be completed within 100-200 days I would be very surprised if the boss was annoyed at the project being completed in 70 days.
While it may literally mean 'from the start of day 100 to the end of day 199' that meaning is unclear and likely to be misinterpreted. A more plausible way to express that idea would be either to use "not sooner than 100 days but no later than 200" as you suggested or, more likely, to give specific dates (e.g. "between November 14th and February 22nd").