J-2 holders can get work authorization and work as per the licensing requirements of their profession.
Contact the Waiver Review branch in USDOS first. Figure out what is going on with your pending case. You cannot apply for more than one type of waiver at the same time.
It is entirely in the discretion of the consular officer whether or not to give you a J-2 visa. Impossible to predict.
If you have neither complied with the J-1 home residency requirement, nor obtained a waiver, you cannot get an H-1. Make sure you are indeed subject to the 2-year HRR. You may not be - then an H-1 is a possibility. Go here to ask DOS for an advisory opinion: http://travel.state.gov/visa/temp/info/info_1288.html
If the consulate is not convinced of your nonimmigrant intent (214(b)), it is extremely difficult to remedy that. Usually, people in that situation should explore options like H-1, L-1, green card - all of which do not require a nonimmigrant intent (intention to remain in USA only for a brief period of time).
If you travel during change of status, the request for COS is deemed to be abandoned.
Assuming you have no home residency requirement, you can start the H-2B process even while you are here in USA.
To the best of my knowledge that information is incorrect. You can file AOS only after the waiver is approved.
I see no reason to leave. The 60-day advice makes sense only if you have just entered USA.
One of the questions I am asked quite frequently is whether or not an E-1/E-2 visa holder can apply for a green card and not jeopardize his or her E status. The answer is PROBABLY yes he can.
In the E visa context, this is what the govt says:
9 FAM 41.51 N15 INTENT TO DEPART UPON TERMINATION OF STATUS