Have old wiring

Ok here's what happened. I was running a air compressor on an outdoor outlet. The circuit breaker blew a couple times, I would flip it back and it was fine. The third time it tripped I reset it but there was no power. It appears as though a few of the kitchen outlets were on the same circuit ( the outdoor outlet I was using is right next to the kitchen) and now now of them work either. Using a non contact tester shows there is power to all the outlets, however if I plug in the little tester that came with it into the outlet I get NO lights on it at all. From what I have seen none of the outlets are equipped with a GFI. Also, all the wires in the breaker box do appear tight and well secured. Any advice or help would be greatly appreciated. This is my first house so I'm trying to learn a thing or 2.

I am trying to change my breaker from 50AMP (double) to 30AMP ( double) - I know the type of breaker - have already purchased from HD. I need to change the laundry socket from 50A to 30A - not sure why it is 50, but I am installing a separate dryer and washing machine...and for the dryer, the manufacturer says to use a 30AMP. My washer will go into a separate socket for 20AMP - which is on a separate breaker. Question 1 I understand that it is okay to 'downgrade' from a 50AMP to 30 AMP but not necessarily the other way, but I wanted to be sure. Please advise. Question 2 I also cannot seem to identify the wire gauge ( no identification that I can see), so I was hoping that someone might be able to identify from the pics and let me know if it is okay to swap the 50 to the 30A at the breaker box, without causing fire potential because of the wiring. Question 3 I also understand that since I have a 3pin plug - that this was code before...but since it is already there - that I can leave it installed this way without having to change to 4 wire. Is that right. Can anybody help me and let me know of any other issues that I might need to be aware of. Thanks in advance for your help