Can paracervical put you to sleep?

Unlikely. Other than taking away your pain which may make you relax and feel better....No.
Maybe. A properly done paracx numbs the uterus and will keep you awake but sedation can be given I.V. Also if too much goes into the vein or too much local is given you can go asleep by seizures or just the drug itself. I, v, local makes a bad sedative.
No. Parcervical block is injection of local anesthetic (numbing medications, not sedatives) around the cervix to facilitate procedures on the uterus such as d&c. You may, however, require sedation to tolerate the injections or discomfort in areas not numbed by the block.
No. A para cervical block is performed using local anesthesia in the pelvic region to facilitate delivery, it is done with the intent of not putting the patient to sleep.
No. No, this is a local anesthetic. It should numb the area but not put you to sleep. Depending on the procedure, sometimes IV medication can be used to relax you, but this is rarely needed.