Will the ER extract a tooth causing severe pain and is broken?

No. Most er do not have staff on hand for "routine " dental care, only for emergencies like trauma.
Doubtful. Most er's do not have a dentist or oral surgeon on staff. Those that do have an oral surgeon on staff may be willing to remove the tooth, but most er doc's will only place you on antibiotics and tell you to see a dentist or an oral surgeon.
I have not seen this. An er is not set up to extract a tooth. There could be exceptions. In my experience the er will help manage tooth pain with pain medication & or antibiotics to minimize infection. They will refer you to your dentist for treatment. If you have an md, they may be able to help & save you a costly er visit. I am available 24 hours a day for my patients' emergencies, maybe your dentist is also.
Most often no. Most er 's will not. More often than not the emergency room will only provide medication for a toothache .
Not likely. If you have signs of infection, you maybe given antibiotic and pain med, then sent home to follow up with your dentist. If no infection, then just some pain meds. So, easiest and cheapest is to call your dentist and be seen. Good luck.