Learn English – ‘within few next days’ OR ‘within next few days’

I see no grammar issues with either of “within 30 days of receiving your application” or “within 30 days after your application is received”. I prefer the phrase “within 30 days of receipt of application” or “within 30 days of receipt of completed application”. The number 30 can be replaced by 90 without affecting grammaticality.

‘within’ and ‘after’ are both performing a similar function and therefore ... should not both be used in the one construction

While at first glance it may seem that using both of within and after is redundant, in point of fact using both words removes an element of ambiguity present in both of the other constructions. Although “within 30 days” is often interpreted as meaning “within 30 days after”, it can also be interpreted as “within 30 days before or after”. Typically, one interpretation or the other is obvious, and after need not be explicitly mentioned. The slight redundancy it introduces is unimportant. In short, add the word after or leave it out as you prefer.

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").