I am a non-EU citizen and I have a 90-day EU spouse Schengen visa. Am I allowed to stay for longer than the 90 days in the Schengen area because I have applied for a resident permit in a Schengen country?
The details depend on the country, but in most cases, an in-country application for a residence permit allows one to remain while the application is pending.
More importantly, if you are the family member of an EU citizen, and you are there in the country with your family member, the penalty for staying for over three months without registering must be proportional, meaning it must be similar to the penalty a citizen of that country would face for a similar omission. You cannot be removed from the country or banned from it. But it seems that you have registered (that is, applied for your residence card), so you are entirely in compliance with the requirements of the directive.
Countries are not required to apply the directive to family members of their own citizens, except in limited cases, although some do. If the country you're in is the country of your spouse's citizenship, the free movement angle might not apply. But the general principle in the first paragraph probably does.