Minimum 3/4" pipe.
Pipe bore is the actual minimum inside diameter of a pipe, which is not necessarily the nominal pipe size.
It should be 3" minimum, preferably 4.
3/4
Depends on how much is flowing into the pipe. One bathroom sink, 1 1/4"
3" pipe usually, but local codes may specify 4".
250 mcm
That would depend on local codes and the fixture demands
Please be more specific about 'service'.
Normally 11/2" DEPENDING on developed length of piping and fittings
50 amps
For a 200amp service you need number 4 copper