This only addresses part of your question, but this question could be answered with a whole book.
Homebrewing Biodiesel
I do want to clarify that biodiesel is not only a potential "industrial" reuse for used cooking oils. Used cooking oil can be processed into biodiesel by a modestly savvy do-it-yourselfer at home, with a simple processor that you can build yourself out of parts you'll find at most hardware stores (plus maybe a pump from an industrial supply store like Harbor Freight in the US).
The complete plans are beyond the scope of this question, but you can find good information online here. Basically, someone who drives a diesel vehicle can turn an electric hot water heater and a 55-gallon (rinse) barrel into a biodiesel processor for cooking oil. The recipe would usually involve adding some methanol, and a catalyst like KOH. You would make batches of about 50 gallons of fuel at a time. Certainly, your own cooking oil would not be enough to provide feedstock for this, but if you added your oil to that collected from local, independently-operated (†) restaurants, you can have a more sustainable way to reuse the oil, compared with consuming gasoline or petro-diesel.
This type of process can supply biodiesel that can be run in most (‡) diesel engine vehicles, with little or no modifications (older diesel vehicles will probably need new synthetic rubber fuel lines if they had natural rubber lines before).
Waste Vegetable Oil (not biodiesel) as Fuel
If you don't want to have to add to your supply with used restaurant oil, another option for people with diesel vehicles is to buy/install a conversion kit that allows you to use (filtered) waste vegetable oil directly in the vehicle. Basically, the car starts on normal diesel or biodiesel. Once the engine comes up to temperature, you flip a switch, and the fuel is then drawn from an auxiliary waste veggie oil (WVO) tank. You normally would switch back to diesel/biodiesel before shutting down just to purge the lines.
The number of diesel vehicles eligible for WVO conversions is a bit smaller, though, and as opposed to using biodiesel, which has superior cleansing properties to petro-diesel, using WVO needs to be done more carefully, because you can gunk up your engine. Older Mercedes diesels (1970s - 1990s) seem to do the best on WVO, compared to other cars.
Oil Criteria
When you use cooking oil to power a diesel engine, you do want to pick good oils. Generally, that turns out to be the same kinds of oils that are more healthy for people. Veggie oils are better than animal oils. No solid fats. Cooking oil can be reused, and restaurants do reuse it. The more it's reused for cooking, the more rancid and more contaminated with food stuffs, the less desirable it is for fuel feedstock.
Oil Handling
You'll want to collect it in clean metal or HDPE containers (5 gallon "carboys" work well). At home, you need to test your oil before using (for example, to see if one restaurant's product is typically usable). The oil then needs to be filtered with a 5 micron filter. After you process the oil into biodiesel, make sure you have it stored safely. Unlike gasoline, diesel doesn't ignite without high pressures. However, your area may have storage regulations to be considered (for example, I seem to remember that California limited storage to 55 gallons or less of fuel, without a special license).
† Large, chain restaurants will have rigid oil disposal contracts with rendering companies, so don't waste your time with them. Deal directly with smaller restaurants, preferably ones you already go to, to save driving!
‡ Note that newer, "Common Rail" diesel engines with diesel particulate filters may no longer be able to run rich biodiesel blends; ironically, partly because of new emissions requirements (that are driven primarily by petro-diesel exhaust!). So, do some research (or ask another question) about biodiesel blends appropriate for your engine.
