We have been eating crayfish for a long time; indigenous French (or European?) crayfish were delicious but they have been wiped out by this species.
There are various streams near me where you can go out in summer with a piece of bacon, some string and a box (maybe a net too) and catch signal crayfish. You don't need a permit as they're an invasive species - in fact you're not allowed to put them back if you catch any.
(But do put them back if you catch the native species! The signal crayfish are larger and have big red claws with white flashes which give them their name.)
edit: was writing my now largely redundant post as masklinn posted. I hadn't realised the Marmorkrebs were smaller - does that make them less of a threat? It does say they're already a problem for native species in Madagascar.
I had thought you only needed the licence to set up traps, which I'd rather not myself as there's a risk of catching/injuring/killing other species, particularly native crayfish or otters. Sorry for the inaccurate post. Sadly it's too late to edit it or I'd put strikethrough all over it...
More details about catching (mostly with traps) and licence details below. http://foragedfoods.co.uk/how-to-catch-signal-crayfish/ https://www.gov.uk/guidance/permission-to-trap-crayfish-eels...
To answer your questions, I'd wait until at least spring; I've only been in summer. Not sure about speed or depth (I've seen them in fast shallow streams and in a slower metre-deep river) but they like lots of hiding places like rocks or overhangs.
AFAIK indigenous european species have mostly been displaced by various american species (mostly spinycheek, signal, and louisiana crawfishes) — and severely hit by diseases those brought[0] — over the course of the 20th century. The marbled crayfish is a recent development.
I don't think the marbled crayfish is great cooking fare, it's pretty small.
[0] especially the crayfish plague: https://en.wikipedia.org/wiki/Crayfish_plague