If I have a syringe and some lidocaine, can I numb my own broken tooth?

Not recommended. Self medication and use of local anesthetic is not recommended as you can have serious repercussions. Better, call a dentist for examination, x-rays and treatment. Even if a pain reliever alleviates your symptoms, you should still see a dentist for evaluation and treatment.
DIY Dentistry. Not advisable. Even if you're lucky enough to avoid a blood vessel or a nerve, do you really feel qualified to extracted a rotten, festering tooth? Face it, you need to see a dentist. Make the call.
No. Unless you are an incredibly brave and stupid dentist or anesthesiologist, I wouldn't advise self-anesthesia. First, results only temporary. Second, if you do it even a little wrong, you could cause numbing of your throat, causing you to choke and aspirate on your own spit and possibly die. You need attention to the tooth, then the pain will go away.
Not advisable. Even if you have a syringe (assuming it is sterilized) and Lidocaine you probably shouldn't numb your own tooth. If you are in pain or need to get your tooth fixed you should visit your dentist. By numbing your own tooth, it would only be a temporary relief from any pain (2-3 hours).
No. If you are not trained in giving oral ijections don't do it. What would you do if you did numb it. Go to the dentist.
Should not. I would not advise this as it could be lethal if you inject into a blood vessel.