For starters, Jason and I (in the recent past) have confabulated on this subject. We both agree that you should check w/your local municipality first to see if there are any 'restrictions'. Secondly, we agree that you should wear long sleeve/pant clothes, at least a dust mask, and you should remove the shingles so they DON'T become friable. These shingles are considered non-friable which means the asbestos will pose a direct hazard if it becomes airborne. Assuming you don't go busting them up all over the place (and you keep your hands away from your face and such), it's safe to remove yourself. BUT, again, check w/your Municipality first. In the overwhelming majority of Municipalities, the restrictions are very limited. Most have to do w/the method of removal and the disposal.
Now, RE: disposal - You contact a Hauler that is licensed to remove a hazardous material such as this. Get this info in writing (such as requesting a copy of the license), and be sure your contract specifically states PROPER disposal of the material.
Regarding painting or residing over this material - Yes, you can paint it. You should wash it first, prime it, then paint it w/a Latex-based paint designed for these shingles. You LOCAL Paint Retailer can help you. This paint isn't cheap BUT it is VERY durable. Don't skimp on paint because as you will find out (assuming you paint it), that you DON'T want to revisit this job for a very, VERY long time. (You get what you pay for ...) RE: Residing over it, I HIGHLY recommend AGAINST it. Most Mfgrs. of any siding will state in their Warranty that you will VOID if if the new siding isn't installed over a SMOOTH, FLAT surface. Also, as the Sider goes nailing up your new siding, what do you think is going to happend to the 'busted' shingle pieces that fall off of the side of the house??? Well, they will collect and 'bulge' the new siding. In short, don't do it.
Assuming there are no major restrictions on removal and disposal, you can do this job yourself. It will be a pain, of sorts, especially if you're doing it w/just a ladder. The use of rented scaffolding or ladder-jacks is highly recommended, and well worth the cost. You simply stack the material in the driveway/yard, and when you're finished, you THEN call your hauler to bring the dumpster. (FIrst, find a hauler before you begin the job. The reason you don't want the dumpster on your lot as you do the job is because 1) your neighbors will fill it at night, and 2) you pay for rental a week at a time. By NOT having the dumpster delivered in advance, you avoid the neighbor-problem and potentially extra fees.)
While you're at it, get ALL your 'junk' that you want to get rid of too. Just tell the hauler you have things like stumps, cement blocks, old bed frames, insulation, roof shingles, and so on, to put in the dumpster. You'll need to tell him how many 'squares' of siding you have. 1 square = 100 sq. ft. of siding. (Just do the math around the whole house.)