The most important step to applying is to begin talking to a recruiter the summer or fall of the year before you plan to matriculate. It's a long process with 2 interviews, MEPS physical, and A LOT of paperwork (which includes a rather brief essay of why you want to join- basically a modified personal statement).
Your payback will be year for year, so for every year that the Navy pays your tuition you must serve one year of active duty. So if you do the 4 yr scholarship then you must serve active duty for 4 years, and the 3 year scholarship will require 3 years of active duty service. You should also be aware that after your active duty time is complete, you are required to be on either reserve or Inactive Ready Reserve (IRR) for 4 additional years. Reserve requires one weekend a month and 2 weeks a year time commitment, while IRR is just one weekend a year. It is highly unlikely that you would get called to serve unless something terrible happens. IRR would be the best option if you just want to get out after your 4 years of active duty, since it only requires you to do "Navy stuff" 1 or 2 days out of the year.
The oral surgeon thing is not true. A dental student who wants to go into OMFS would graduate from dental school, complete their 4-6 yr residency and then go on active duty for 4 years.
As far as the financial aspect goes, they pay for 100% tuition, books, and equipment. There are occasionally nominal fees that are not covered. During your time at school you will be paid a $2200/mo stipend for 10.5 months. For the other 1.5 months you will go on ADT (active duty tour), but don't freak out. ADT just means that you get paid as an active duty O1 which will essentially double your monthly salary. You have the option to do one of three things on ADT: go to Officer development school (ODS), do a clerkship at a Navy dental clinic, or you can take school orders. School orders means you just stay at school and carry on as you normally would, but you cannot go more than 350 miles away from school. Nothing else is required of you. If you're lazy then that is the best option Lol, but at some point you will have to go to ODS before active duty.
If you do commit for 4 years, then you will also receive a $20k signing bonus (taxed at 25% thanks to uncle sam). Your military pay as a dental officer will be 90-95k depending on where you live. Hope that helps and good luck!