Can 10-3 wire be run between in my basement between a transfer switch and the outdoor receptacle which is connected to a 30 amp 120/240 vac outdoor receptacle from a potable gnerator plug?

You need a 30 amp power inlet box, 10-3 is 4 conductors, and is what you need. 10-4 would be 5 conductors and not needed.

30 amp power inlet box - Google Search

Your question is very confusing, do not feed anything off your gen. connection. If you have 2 devices that may use the same wire and your trying to kill 2 birds with 1 stone, your taking the lazy way out. 1 dedicated line for each. You don't want a break between devices. More problems can arise down the road... Alway's do it right the first time...