First of all, I have no experience with lasers.
Is there a cheap laser (~150EUR) that can cut 10cm thick XPS?
AFAIK, there are two types of lasers: laser-diodes and real-lasers.
I guess real-lasers would cut XPS with no problem but are expensive.
Are laser-diodes strong enough to cut XPS?
I understand that laser-diodes are better suited for engraving,
because of focus problems, right? If I would somehow change the
focus dynamically, would it then cut 10cm of XPS? At what speed?
Short answer: No.
It would be a challenge for any laser to cut 10cm (4 inches) of XPS foam.
A CO2 laser and correct optics would work, but you won't get a parallel cut.
Hot wire is the best method for that thickness of foam, and much cheaper too.