Jimmy, first you need to get your termanology correct. Romex is a trade name. There is no outdoor rated Romex. Art.334 for NM cable read its permitted and not permitted uses. then look at Art. 340 for underground cable.
You did not state the voltage of the water pump so lets say it is 240v the amps would be around 12amp. If your longest run is indoors then you can run #10 NM (Romex) to a junction box then change to #10 UF to make the transition to underground direct buried. On a 20amp breaker. The only thing the #10 will do is keep you from having too much voltage drop.
You need to supply more info on the welder such as duty cycle.
As Dennis said if these two are going to the same structure then it is a violation.