No. A household receptacle is 120V. It will not work.
Either there is something drawing a lot more current, or the breaker is faulty. You may need to have an electrician check the breaker for you, or if you feel competent, replace the breaker with another of equal value. That would be a lot less expensive than having an electrician drop by. BUT... The startup current of a household freezer AND the "small pump" may add up to enough current that it's tripping the breaker. Read the labels and see what the actual current draw is on the pump AND on the freezer. If the total current approaches the value of the breaker it might be best to move something to a different circuit.
The sump pump breaker alarm will keep going off if it is being overloading. This is usually a result of a failing pump or a clogged filter.
Normally AWG #10 wire on a 30 amp breaker but it really depends on the size of the pump. The pump manufacture will specify the size wire and size breaker needed.
no
The sizing of the circuit breaker and hence the wire size to feed the pump are all dependant upon the amperage of the pump. This question can not be answered without that information.
No.
Depends entirely on the voltage and current rating of your pump.
My heat pump was professionally wired when built in 2004. Wire is No. 4 AWG with 80 amp breaker
A breaker is based on the amperage that is drawn by the pump motor load. Find the full load amperage of the motor. The wire fed from the breaker has to be rated at 125% of the motors full load amperage. The breaker for motors have to be over sized, usually 250% of the motors full load amps.
Sounds like a 240 volt sub panel feeding a well pump.
Not advisible.
No, they each need their own breaker of the right amp. Neither of them would cause the breaker to trip if there was a problem.