16 inch bar works great on the 310.. or any high powered saw. No one says to have to run a long bar... I run a short bar most of the time I'm doing firewood, even on my 361, and mount longer bars as required. Less weigh and great cutting.
As for the pump, yes, definitely check the chain tension etc. The 310 has the higher output pump (of that series saw) and should never smoke on the bar. Take off the bar and side cover, start the saw and see if oil pours out of the small hoke at the TOP between the bar studs. Not a "splutter, but a nice constant stream at medium RPM. If it does, look at the tiny oil hole in the bar and make sure it isn't blocked. If you've turned the bar over and not cleaned out the hole, it's blocked.