Is it possible to fit a 10.5 cm mug? Thanks in advance

8 cm and higher, you can remove the drip tray with a flick of the wrist.

Since not everyone has the same or the same amount (;-)) of cups in the cupboard. Can the height be given in mm or cm, for the size between the plunger and the tray ?

Only espresso cups. You can, however, remove the drip tray so that larger cups also fit underneath. But the machine is also mainly intended for espressos.