OEM type filters shold be DRY. If its a K&N type, then yes, it should be oiled to capture dust.

Back to OEM, if its a pleated filter in the airbox and covered with oil, you have another problem.

Your running issue (aside from a possibly clogged air filter) could also be the throttle position sensor.

Just for reference, the filter is a K&N.

Another quirk is that the engine just turns over the first time the key is turned, and then starts up perfectly on the second turn.

It sounds like the check valve for the fuel pump is leaking. Try just turning the key on for 5-10 seconds and listen to the fuel pump...and then try starting. If this works, its soon gonna be time for a new pump.

x2, when mine was doing this it could have a rough idle for a few seconds before it smoothed out because of the air in the fuel system and had to push the air out an da cyl or two would run a little rough. pump fixed it for me... put a gauge on it and watch for the preusse to bleed off.