What is the difference in procedures for bleeding different disc brakes?
It's necessary to understand the design of the systems to understand why Avid requires a separate bleed kit. 

There is a significant difference in the design of the brake system between Avid and other manufacturers such as Shimano and this is the main reason on why Avid have a proprietary bleed kit. 

Although an 'open' hydraulic system Avid's Taperbore resembles a 'closed' system, one that operates in a vacuum. The Avid system uses a small reservoir fixed amount of fluid in the system and a bladder around the master piston with air in it. Shimano and other brands use a larger reservoir with a bladder separating the brake fluid (in this case mineral oil) and air. In both cases air is used to compensate for the fluid in the system expanding.

Because the Avid system operates in a vacuum it requires all air to be removed from the system and that is why you need the two syringe set, other designs simply require fluid to be topped up in the reservoir and air worked out.

All the different brake fluids have different characteristics (boiling points, freezing points, amount of expansion etc.). Avid specifies DOT 4 and 5.1, which is a pain. Like all solvents you don't want to get it on your hands but it isn't going to kill you unless you work with it everyday without protection. DOT fluid is hygroscopic, it attracts water so although you can buy it in larger quantities cheaper than through Avid it doesn't have a good shelf life once opened and probably shouldn't be opened again.

To bleed Elixirs you don't need a Avid bleed kit, just a hose barb that fits and some generic hose. I use the brass hose barb which is for my Rockshox x-loc and generic hose and syringes.

Also you can do a less complicated 'speed bleed' to remove air from the system, but a full bleed is better. I bleed the brakes on my hack/ commuter but leave it up to the wrench at my LBS to bleed the brakes on my XC bike (I just don't see the point in paying a few hundred for good brakes and then not getting them optimised by a professional).