Had an impacted and broken tooth pulled 2 months ago, but still in pain is this normal?

It is not normal. Pain that lasts for up to a week, but is gradually getting better is normal. Pain that starts to get worse or lasts more than 2 weeks is considered abnormal and you may want to see your dentist. Even if you had wisdom teeth removed couple of months ago, there could easily be something else going on at a different part of your mouth/teeth and examination is warranted.
No. See your doctor to have the extraction site looked at. Sometimes there may be an adjacent tooth with a cavity that is causing the pain.
ROOT BROKEN--- NO. The broken root should be taken out or it may cause infection or abscess ( foreign body) in the gum.