You're required to file for SSDI due to your LTD policy, correct?

It'll be difficult to be approved prior to having a diagnosis, but by the time you end up at an ALJ hearing, hopefully you'll have that diagnosis.

Most insurance companies will cover the cost of your representation. They often prefer that you use the service they choose, but if you read the fine print of your policy, you can generally hire whomever you wish. If you go with an attorney, you don't really need to sign until the ALJ stage. If you go with a non attorney rep they will often be more hands on from the start, so you could sign with them as early as you wish.

You need to be aware that the standard is if you can work at the SGA level(Significant Gainful Activity is $1130 per month in 2016) performing any job, not just your prior work. So you'll need to prove you're incapable of even unskilled, sedentary work.