Do you ever need to revisit gathering or mission skill trainers?

Nope. Gathering and Mission skills are set and forget. You'll gain access to new missions by leveling, or from items you can find or buy on the GTN. There's no need to return to these trainers.

As far as I've been able to determine, no.

Gathering nodes only require a certain number of skill points to gather.

Missions are automatically unlocked as you increase your skill points. Note that you can still select the lower tier missions through the tier dropdown in the upper-right.

Special missions can be gotten through the Slicing profession. These one-time only missions have a much higher payout than normal.

While the answer from @LessPop_MoreFizz was correct now with 3.0 and also back in 2.0 to get advanced or expert crewskill missions you have to go back to your crew skill trainer to buy the increased crew skill level.

Advanced crew skill - 10.000 credits - level increase to 450 

Expert crew skill - 50.000 credits - level increase to 500