What's the proper english name for this mechanical tool to remove bolts of the door?
I've always heard it called by the originator's brand name: a Yankee screwdriver.
The term "Yankee screwdriver" is often used to describe push/pull type screwdriver other than one manufactured by North Brothers Mfg. Co. or Stanley Tools, who purchased the rights to the well-known Yankee brand or trade name in the 1940s from North Brothers. North Brothers always marked the tools they manufactured with the Yankee name, and in most cases the North Bros. name and location as well1. Text and pic from Wikipedia
By the way, my brother is a carpenter and that's what he calls those things too, and although he mostly uses power tools these days he still keeps a couple of these around.
Additional note: I've also heard them called just by the brand name: "Hey, pass me the Yankee."
This one says "semi automatic mechanical screwdriver"