Hello, hope it's not too late.
1) Since it's in the same country UAE, the Visa fee should be same(?) though, there seems to be little bit difference between Al Wijajah (Hatta) border and other borders in other Emirate (such as Abu Dhabi(Al Ain belongs to Abu Dhabi Emirate), Ras Al Khaimah, Fujairah).
As for crossing the border, frankly, from my experience, it depends on your luck, or I should say depending on the person in charge of the day.
As far as you have valid visa coming from Dubai Emirate or other Emirate, you should be able to get out from other border though, sometimes "the person in charge of the day(officer?)" stops you and order you to return to the border where you've entered(!!). This is true and I don't know what is the story. In care if he/they force you (by oral) to return to the same border which you've entered, you have to follow their order and there's no other way. It happens often, not only me (I faced only once)...it's true.
2) Yes, you can reach upto "the entrance" of Hilli border from Al Ain by taxi, and usually most(or all) taxi from Al Ain cannot take you upto Omani border. So what you need to do is to get off from the taxi, after paying the charge. Then walk into the Hilli border, ask the staff where you should go to get "exit" stamp from this border. After getting the stamp, walk out from the border(HIlli check point), then walk a little bit, you're already in Al Buraimi, Oman. And catch a taxi, which you can take you directly to the border(Wadi Al Jizi check point, which is border of Oman side), otherwise ask the driver to take you to the taxi stand where you can get another taxi which will take you to the border. It takes about 30 minutes one way.
3) As you may know?, if you come from Hilli border, there's ONTC terminal near there, which is nearby TOYOTA showroom. Bus will leave from there for Ruwi, Muscat at 13:00 and 17:00
ONTC Buraimi office Tel.: 25652424
Good luck :)