Garden Spade

Yes, it is called a cut the grass.

Yes, it's a grass trimmer, http://www.hirestation.co.uk/hirestation ...

... Is a rental company. Others, just search "Turf Cutter hire", etc. ..

Yes, this is called a grass cutting. You can usually rent for rental.

If you want to cut back lawn topsoil, a cutter. If you want to dig around for other purposes, a rotivator. I rotivated my lawn side, then re-sown grass - rotivator to sign a rental company's tool room.
Horses for courses.