What is this green micro-organism found living in water?
I can't decide if it's a eukaryote or a prokaryote.


Well, that's certainly an eukaryote. We can clearly see the nuclei in the cells of your fist picture, as well as in the cells of your second one (not so clearly, though):



In the image above, the arrow starts right on one of the nucleus of that group of (apparently) three cells. 

Besides that, despite some cyanobacteria being very long, 10&micro;m is a lot for a prokaryote. The mean size of cyanobacteria is way less than that.

By the way, this:


  ... a prokaryote or a cyanobacteria (that can combine both characteristics)


Doesn't make much sense, because cyanobacteria are prokaryotes. More precisely, they are Eubacteria prokaryotes.

Finally, what eukaryoyte is this? I can't tell. It's most probably a kind of Chlorophyta, but I'm not able to go to the Class level. If you're interested in identifying it, you can try a taxonomic key. This one is for British freshwater algal flora, I don't know how different French algae are from the british ones: The Freshwater Algal Flora of the British Isles: An Identification Guide to Freshwater and Terrestrial Algae.

If I'd guess (and this is a wild guess), I'd say that this is a Chlorellales, as the famous Chlorella:



Also, have in mind that saying "that's an alga" doesn't help as well: worse than paraphyletic, algae  is a polyphyletic group and, in biology, polyphyletic groups mean almost nothing.