Confusion about how an electron gun works

I cannot comment on the construction of a specific electron gun, but in a typical CRT, the internal surface of the tube around the screen is coated by graphite, forming the final anode electrode.

This electrode is connected to high positive voltage and one of its functions is to collect electrons arriving to the screen.

So you are absolutely right - any cathode ray tube or electron gun should have some return path for the electrons. It is not shown on your diagram, probably, because this diagram is simplified.

This is an Electrical Engineering question. The target is usually not in the hole (opening) of the anode. This is because, first, the hole is small. Second, you often want to be able to manipulate the electron beam, like what people do in the CRT TV. So the target is usually at the right end of your diagram, where your blue arrow points to.

The target is usually connected to the anode so its potential is equal to the anode. You can either connect the anode (and your target) or the cathode to ground. In a microwave oven, the anode of the magnetron (a kind of vacuum tube with anode, cathode) is grounded. In a TV's CRT, the anode is at high positive voltage and other component (maybe the cathode) is grounded.

This diagram might make it clear to you,