Can the main circuit overheat causing the buttons
To trip as our main circuit breaker was very old and tripping? Could the brand new washer circuit contribute electrically to the tripping of the old main circuit breaker such as the new overriding the old? Thank you for educating me.

We had a dryer electrical fire on fire on our
Gas dryer in 2011. A new washer circuit was installed. We have had recent electrical problem to where the buttons on the old main circuit breaker kept tripping and shutting off our air conditioning and electric garage door. We recently got a brand new circuit breaker panel. This happened, now, one year later. Do these problems, any way, connect with the fact that the washer breaker and a new main circuit breaker was not installed at the same time on year
2011. How could I gove you money for your appreciated time and effective help? I

When breakers trip as you say they do it is a sign you need to have a PROFESSIONAL on site survey done on your entire electrical system to determine why they are tripping to start with. This can be a hazardous/unsafe situation that cannot be corrected without actually seeing and testing the circuits in your home. I would really like to help but I cannot do this work on this message board so I have given you the best advice I can. CALL for a local professional electrician to examine your electric system before more problems show up. Sorry this is the best I can do from here.

Can the main circuit overheat causing the buttons
to trip as our main circuit breaker was very old and tripping?
YES if the service entrance conductors are to small for the draw or damaged. The wiring could not be tight in the connections etc.

Could the brand new washer curcuit contribute electrically to the tripping of the old main circuit breaker such as the new overriding the old?
Not really since the main breaker is much larger amp wise than the washer circuit.

Time for you to call for some local on site professional help for this problem before it causes more serious problems.

NOTE: many times people install a new circuit breaker panel that is to large for the wiring from the utility into the home. They do this thinking that a new more powerful circuit breaker panel will give them more power to the home. This is false since you also need to change the wires going into the new panel from the utility outside. The wire size needs to match the capacity of the new panel. This can and will create problems just like you are having and it is a serious hazard.

Over time the screws holding the wires on the breakers loosen from heating and cooling many times over and over. The wires will get so loose that it either causes a fire or heats up the breaker enough to trip it. You can also see where the plastic coatings on the wires burns away. This happens in receptacles and on breakers, so it is a good idea to go around the house and tighten all of the screws on the breakers and on the electrical receptacles replacing them when they have burned or cracked. Don't neglect the light switches. A house that is old is a real fire hazard as the insulation on the wires burns off for a foot or two into the wall. This is how houses catch on fire. If your lights flicker it is a sign that you need to act before the house burns down. Since you have already had a problem it is a sign you need to take care of this and check all connections unless you are trying to collect on fire insurance.