Now some of it's shoots are a few meters long and some of them went back into the ground and rooted there quite deeply.

I know that a new bush will grow from that point but I don't know if I should cut the cord between the mother bush and the new one, when to do it, and where to to it.

Thanks!

I'm in Israel, in the Sharon area (7 km from the beach, very close to sea level)
January here is 14 degrees on average, but this has been the driest January ever measured!
Yes, they are purplish-black.
I now know about cutting to the ground after fruiting, but that wasn't my question.

Also there are very few leaves and they don't look so good - there is lots of browning at the tips.

Yup, end of season when fruiting is done you might as well cut the spent fluorocanes away so they stop blocking off sunlight and consuming stores in the roots to increase yield next year's fluorocanes.

And it sounds like it is invasive in your area so you must prune to control spread (size).

There are several types of raspberry and hybrids. There may be primocane fruiting types now as well--but they do not do well in my area...no raspberry really does--so I don't keep up with developments. I imagine most parts of Israel are not ideal for raspberry development and much partial shade particularly in the afternoon is needed.

commercial guide:
Commercial Everbearing Red Raspberry
Production for New Mexico
http://aces.nmsu.edu/pubs/_h/h-318.pdf

So how do I know if the plant is dormant?
How do I know if it's sick or lacks irrigation?

"How do I know if it's sick or lacks irrigation?"

Experience! Water more, see what happens. If it responds well you figured it out and will know what to do next time. If it doesn't well that is some knowledge too.
Tips browning is usually caused by damaged roots, damaged roots can be caused by lack of water, too much fertilizer, not enough fertilizer. A mineral deficiency (use trace minerals). Insect infestation like nematodes, or cane borers. Fungal infections seems unlikely in your dry conditions.
Increase water, inspect canes for lesions, if it dies see if roots have tumors (nematodes cause this). Add green sand or azomite or kelp to increase trace minerals. This might not be needed, but if nothing else works, do it.
Add compost to top and mulch with something every spring or at the start of new growth once a year.
I add composted manure on top and also add a touch of organic fertilizer to the compost. Then I top with pine needles as a mulch. This is ideal IMHO, but it needs the basics, food and water. Worst case, just water and miracle grow. Better than nothing.