I would suggest the easiest way to cook ground beef with only disposable items is a grill. It's worked great for more than a couple of years.
But... If you don't have that, there are a few other things you can do.
Labor intensive but extremely effective is the raised rack made out of foil. You construct a tray out of tinfoil and add ridges to the bottom every inch to an inch and a half. This will cook your beef and allow grease runoff so you meat doesn't just cook in it's own fat.
The only other thing I could think of would be to cover one of your burners completely with tinfoil and use it like a frying pan, but i'm prertty sure that's gonna result in a lot of cleaning time, so it would probably be a negative once you took it all together.
Smoke is normal in an electric oven, but flames are definitely not.
In order to start a fire, you either need a spark, or you need to heat something beyond its autoignition temperature (AKA kindling point). You might have had a short - or you might actually be using a gas oven with spark ignition - but I'm guessing your issue was the latter.
Cooking oil or grease being heated beyond its autoignition point is one of the most common causes of kitchen fires (grease fires). Supposedly, some oils have autoignition points as low as 550° F (or 288° C), though I'm not sure which oils those are. Olive oil would be my guess as the lowest, but pepperoni grease could very well have ignited at self-cleaning temperatures (which, as you noticed, go up to nearly 1000° F).
Fortunately for you, all modern ovens have a mechanical interlock which prevents them from being opened during a self-cleaning cycle. If you'd opened it, you would have made the problem a lot worse by (a) supplying the fire with abundant oxygen, and (b) drawing all the hot air and flames out of the oven and into your kitchen, quite possibly setting your whole home on fire. Heat wants to move to where it's cold; that's why you keep your doors and windows closed in the winter.
There are a multitude of oven cleaners available for self-cleaning ovens - you are supposed to use these before you run a self-cleaning cycle. Yes, I know it's odd, but "self-cleaning" doesn't really actually mean that it cleans itself, it just gives you a little extra help. You need to try to clear out all the grease and big chunks of food first using one of these cleaners, then run the self-cleaning cycle to deal with anything you might have missed.
Best Answer
That looks a lot like melted aluminum.
I'd use a solution of sodium hydroxide to dissolve it; probably won't even hurt the oven paint.
Easy Off Heavy Duty Oven Cleaner is NaOH based, so that'll probably work for you, and is easier to find than straight lye nowadays. Just follow the directions. If the aluminum is thick, you may have to do things twice.
Other brands of oven cleaner will likely work as well as long as they contain sodium hydroxide (lye). It will help A LOT for you people who need aluminum foil off your oven!