Re: electrical outlets burning
There are at least a dozen things you should check, make sure you are very comfortable with electrical work, here are a few that you may have thought of yourself.
First - the nasty one, do you have a Federal Pacific panel? If yes, you may want to budget for a major investment and overhaul. See Fifth
Second - do you know the path of all of your circuits? If yes, start close to the panel and work your way out to each outlet to find the issue
Third - you need at least one more circuit. I think it is not recommended to have more than 8 receptacles on a 20a circuit, but consult your codebook, not some putz on the internet. The codebook will also tell you the microwave oven should have it's own circuit.
Fourth - Replace the recpatcle with properly grounded and GFCI protected ones if you havent already
Fifth - If you are not extraordinarily comfortable with any of the above, ask your neighbors, family members and realtor for recommendations for an electrician and then get ready to pay some money. It stinks, but doing the work wrong or having the electrician redo it after you do it once isn't worth it.
It's this old house, not this built after your dad was born house.