I got lidocaine at my dentist and my hands started shaking. Later I looked at my lip and it was dropping will it go away and is this normal?

Yes to both. The dropping wqas due to the anesthetic and the shaking is due to the Epinephrine in the anesthesia. I prefer to use Carbocaine without Epinephrine unless a treatment will take a long time thqat reduces the shaking to zero since there is no epinephrine.
It happens. Mostly from anesthesia given for lower teeth. It typically resolves completely within a few hours. It can be scary but really is more common than one thinks. Next time ask your dentist to use an anesthetic without epinephrine.
Yes and yes. The "drooping" is just because your lips and face are anesthetized. It will wear off in a matter of hours. The Lidocaine anesthetic has Epinephrine in it, and sometimes when a larger quantity of epi enters your blood stream, your heart will race and you may shake. The effect is transient and it passes uneventfully.
Yes. Lidocaine has Epinephrine added to it to make it work better. When you get a jump of epi into your system it will cause the shaking. Many people are sensitive to this, but it is very temporary. The droopy lip should go away in 4-6 hours. If it lasts more than 12 hrs call your doctor immediately.