I'm sorry you're having such a struggle to keep working.
Neither the Americans with Disabilities Act nor the Family Medical Leave Act are any sort of job guarantee, or job insurance. And in order to evoke -- use -- either of them, you have some very specific requirements you must fill. And yes, you can be fired from your job, even though you do have a disability, and even though you have disclosed that disability, and requested "reasonable accommodation".
Here are the legal steps you must take to evoke the ADA. You have to disclosed to your boss that you have a disability, and as such, are requesting a "reasonable accommodation" to that disablity. And you
have to provide your employer with a written statement about what that "reasonable accommodation" might be: fewer hours (yet you'd still do the same work), or shorter hours spread out over more days, or doing the work from a sitting position, or ..
whatever might work for you. Once you've disclosed that you have a disability, and asked for "reasonable accommodation"
(I put that in quotes because it is a specific legal phrase that carries with it particular legal requirements)
the boss has to consider -- note I said consider, not accept --
those "reasonable accommodations". If he (or upper management) thinks you can do the same level of work with those accommodations, then they can let you do that. If they don't, then you can be fired.
You are telling us that you have a certain number of sick days, and a certain procedure you must follow to take those days. Unless your disability somehow interfers with your ability to "follow the rules" and take the steps necessary to use the sick days (calling in, calling someone else to enter it into the computer), you can be challenged about why you didn't do that. It's an expectation of all employees, and the ADA doesn't excuse you from all employee expectations -- only those that you and your employer have agreed upon...in writing. As for taking a week of sick leave "in advance" -- whether this is acceptable or not would depend entirely on your employee benefits manual, or procedure. Most companies cannot afford to have an employee gone for very long without it becoming an inconvenience for a lot of others -- scheduling, filling in for your absence, all that. Most employers would question you being gone for a week, with or without a "doctors excuse".
If indeed you can't perform your job without taking more sick leave than is alloted to other employees; if indeed you can't perform your job, even with "reasonable accommodation" agreed to, in writing, between you and your employer -- then yes, you can be fired.
As for the Family Medical Leave Act -- you can use up those days (I've forgotten what the number is) for your own medical needs, but you need to know that they are unpaid. and once they're gone, they're gone.
I don't know what kind of work you do. You might start looking around your place of employment to see if perhaps there are other jobs you can do more effectively, and without missing so much work,
or if indeed you might have to consider quitting, or working only part-time, or applying for SSDI, or some other alternative.
And no, they don't have any obligation to be concerned, or to make allowances for you, because you are disabled. The whole point of the ADA is to demand that employers treat people with disabilities the same as employees who don't have disabilities...not differently.
(No, I'm not a lawyer. But I have been a disabilities consultant, primarily with people with developmental disabilities, for the past 30+ years...)