Crafting trainers provide you with new patterns for gear or other items as you level up, but do you ever need to revisit gathering or mission crew 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.