No, it should not but if the crown touches the gum, a small amount of local anesthesia just around the implant will make it comfortable.

Rarely.  But sometimes the gums have to get used to a tighter feel because the implant crown is generally slightly wider than the healing cap. Out of hundreds of cases, only one or two patients have needed local anesthesia.  Any discomfort is mild and usually dissipates in a few minutes.  Nothing to worry about.

Placing a crown on an implant should not hurt. If there is any pressure on the gum as the abutment and crown are seated, speak to your dentist. Numbing gel is almost always enough for those few cases where there is pressure.

No

Hi,

If your gum tissue has been properly managed during the implant placement
and healing process, taking an impression and placing the crown should not
hurt. If the implant is buried and needs to be exposed, then there will be
some slight discomfort during the exposing process.