Assuming you just plan to run outdoors above ground get a 10 AWG cord specified for outdoor use. It will be a stranded conductor rather than solid.
Use only AWG # 10 copper.
In North America the smallest size breaker found in a home distribution panel is rated at 15 amps
The biggest circuit breaker in any home is the main breaker located in your main breaker panel that is installed where your electrical service cable comes into your home.
Yes the wire size is larger for that size breaker but will not effect the 30 amp breaker protection of that circuit.
To determine the amps of a breaker box, look at the main breaker or label inside the panel. The number on the main breaker indicates the maximum amperage that the breaker box can handle. Make sure not to exceed this amperage to prevent electrical hazards.
AWG # 10 wire.
Use only AWG # 10 copper.
In North America the smallest size breaker found in a home distribution panel is rated at 15 amps
The biggest circuit breaker in any home is the main breaker located in your main breaker panel that is installed where your electrical service cable comes into your home.
A 100-A sub-panel would be fed from a 100-A breaker.
Yes the wire size is larger for that size breaker but will not effect the 30 amp breaker protection of that circuit.
50 amps
Depends on the wire size you are using. If the generator breaker is a 30 amp then install a 30 amp breaker.
You can reduce the breaker size because that limits the current that can reach the main panel. There is no safety issue other than the significant danger in an unskilled person changing out a main breaker.
50 amps
You can. Using these size breakers in a 60 amp sub panel might be pushing the limit for breaking the sub panels feeder breaker. The breaker feeding the sub panel will have to be no bigger than 60 amps because of the main bus bar capacity of the sub panel. It would be better to install a 100 amp sub panel and then there would be a bit of a buffer and you will have the ability to add additional small load circuits.
To determine the amps of a breaker box, look at the main breaker or label inside the panel. The number on the main breaker indicates the maximum amperage that the breaker box can handle. Make sure not to exceed this amperage to prevent electrical hazards.