You need a 60 amp breaker.
6 AWG wire
The nearest wire size is #6 AWG which is .162023.
4 Gauge wire is larger in diameter than 6 Gauge wire.
It is unusual to have multiple outlets on a 30 Amp service. If you do this you need outlets rated at 30 amps and 10 AWG wire.
minimum gauge is 6 and the best is 4
The size of the wire is stated by its gauge under American Wire Gauge. Six gauge wire is size 6 AWG.
Use AWG # 6 wire.
6
6 AWG wire
It should take a two pole (220V) 50 amp breaker and matching plug wired with # 8 or # 6 gauge wire. Check your manual.
The nearest wire size is #6 AWG which is .162023.
50 amp breaker wired with AWG # 6 wire.
6 gauge
You could if it fit properly in the connector, but it would be massive overkill since you only need 12 AWG wire. You also have to make sure the load can physically accommodate that larger size wire. Every so often someone will replace an electric cooktop witha gas one. All of the new gas cooktops require a 110v outlet. The old electric cooktop may have been run with a 50 amp breaker (and probably 6 gauge wire) In order to convert this 220 v circuit to a 110v circuit we change to a 20amp breaker. Depending upon the brand of breaker sometimes the wire will fit into it. If not, we splice on a short piece of 12 gauge wire and then connect to the breaker.
A 30 amp breaker run 75 feet would require you use AWG #10 wire. But, an oven normally pulls more that 30 amps and I would not use #10 wire on a 30 amp breaker for an oven. Normally any newer oven is wired with AWG #6 gauge wire on a 50 amp circuit breaker so the breaker will not trip during heavy use of the oven. Some older ovens could use a #8 wire on a 40 amp breaker.
Yes, the smaller gauge number, the larger the wire is.
Normally a new kitchen stove will require a 50 amp breaker wired with AWG # 6/3 with ground wire.