Hard to tell From the photo, but might be Ash. If so, it would make a great axe handle. But you can of course make an axe handle out of any hardwood, it will mostly be fine.
Doesn't look like oak to me. For oak, I look for medullary rays that would appear running perpendicular to the rings on the end grain. I don't see any of those.
Ash or white oak would be my first guess, either would be great for an axe handle. Really all your looking for with handles is relatively straight tight grain structure, everything else is more aesthetic than functional
I don't think it is ash - the bark does not look right. Having a cross section showing the face grain and end grain is often helpful to identify wood from a picture.
Yes, those terms match your pictures.
My vote is for beech, but it's still a guess. It grows in the region (Jura Mountains), is a hardwood and has a similar grain pattern and bark.
I loath getting on my high horse, but don't take things from forests unless they are explicitly there for the taking or you have permission from someone. The odd small item here and there may not seem significant but if everybody did it it would have a significant impact.
Thanks for your concern, couldn't agree more. In that case, that forest is partly private property where I own a bit, and the branch came from there. It's not exploited for its wood and is left mostly in its wild state. I'm about the only guy out there that even go there for a walk once in a while. It's still an item taken from it, but believe me, that forest is well cared for :-)
With the dislaimer that I know nothing about trees in France...Here is some info on what grows there. Of what I see on a quick skim, I'd probably be looking to see if hornbeam or beech has the right kind of bark. It looks kind Alder related to me, but that may not be a useful starting point.