Hi Tribulations, I applied exactly 4 years ago for SSI/SSD. I never had to go in for an appointment at any government offices, but I did have to go to an appointment with a pschologist selected by SS. It wasn't long and the guy was very nice. I didn't have an attorney, and received word in 6 months that I was eligible, and started getting payments and back payments a month later.
SSI is for the times when you are/were indigent and couldn't make your usual income due to your disability. SSD eligibility begins 6 months after the date of disability. It's important to let them know when the disability first caused a problem with your income, as well as when you became completely disabled. So apply for both!
The more records there are showing that you have sought treatment for your disability, and over a long period of time, the better. You will just need to provide the dates/providers/places/addresses (if you have them) of treatment, and SS will find any place that has closed.
It doesn't hurt to periodically call the SS office once your file has been assigned to a particular worker, to find out what records they are still waiting for. That's the thing that most often holds up a determination, so if you know which places records haven't been obtained, there is where you can do something - by calling, writing or visiting (best) their medical records department and getting things moving. Just don't call the SS person too often.
You don't need an attorney at first, usually. Most applications are turned down the first time, so if you do get turned down, you may want to consider doing the appeal on your own, and only hiring a disability attorney if the second application is denied.
Eligibility for medical insurance will depend on your monthly benefit amount. Right now, you should be signed up for Medical Assistance, food "stamps", and cash assistance if your income meets the guidelines. Once you start getting monthly SSD payments, your income may exceed the guidelines and there will be a period in which you have neither (although that may have changed with Obamacare). Medicare eligibility begins 2 years after your SSD eligible date, which is 6 months after your disability date.
Whew! I sure hope you were asking about SSD in the US! Good luck, and I hope this helps. :)