whats the name of the gadget that keeps the door open when getting in and out

I believe it is called a "whichamecallit."

 

Front Door Check Assembly

There are three bolts that secure this to the door and to the body. Looks like you'll need to remove the door's inner trim board to access two bolts and remove the part.

Tightening torque is 49 in.-lb on the two bolts attached to the door (applying Loctite to the bolt threads), and 22 in.-lb on the one bolt attached to the body.

Apply multipurpose grease to the sliding areas of the door check.

 

I am new to the door check problem...but that last instruction...to apply multipurpose grease...I had read somewhere else that doing that RUINS a door check...it will not hold the door open (or at least not nearly as effectively), if the door check is lubricated.

I could be wrong of course, but greasing a part that is supposed to hold something by using friction, intuitively does not sound like a good thing to do.

 

Grease will clog it up inside - avoid. Use WD40 or that white lithium spray (for cars).

Make sure the bolts/screws are all tight too as they can work loose.

The check strap is a part that needs continual attention on a taxi, especially on the popular doors.