I don't see a reason to raise your disability unless it's something that will impact your ability to do the job.  According to the ADA (Americans with Disabilities Act), an employer must make a "reasonable" accommodation, but that accommodation has to come from the employee.  In other words, you need to tell them what they need to do to work around the disability.  And the definition of what's reasonable can be a gray area.  For instance, having to redesign the office layout isn't reasonable.  But installing an ergonomic keyboard is.

Now if you're taking medication that might show up on a drug test, then that's something you might want to mention if they ask you to do a drug test.  This way, they're prepared for a positive result in advance and have a legitimate reason for it.

But if your disability won't impact your ability to do the job, or you'll be off by the time you start, then there's no reason whatsoever to bring it up.  As it gets closer to your getting the job, then you might want to mention it.  As for your being concerned about your job being there, I can relate to that.  I went through this twice--once with the S&L crisis and again with the outsourcing of tech jobs. There's nothing wrong with mentioning that you're concerned about being downsized, especially since they've already done it twice.  And so you're looking for something more stable.  No guarantees, but looking to change fields or companies uneffected by this is understandable.

As for whether a prospective employer will contact your current one, there's no definite yes or no.  Some may, others might not.  And even when you tell them not to contact your current one because you don't want them to know you're looking, I've run into some companies where they've done it anyway.  Whether it was by accident or deliberate, I don't know, but it's happened.  However, I think most will honor your request and not do it until after you've started or are getting much closer to an offer.