Useful elements in all answers so far, I'll just drop a few more suggestions. (Context: started a month ago, daily use since swimming pools and public parks are closed right now so I cannot swim or run.)
There are different materials for the rope, I've seen nylon, leather and steel.
I'd recommend leather, because it's resistant enough to sustain outdoors environments. A thick PVC (perhaps what you referred to as nylon?) rope will also work, but still wear out slightly quicker than leather if used on e.g. concrete or bad/rough asphalt.
For good short descriptions of several ropes in different materials, translate the rope products page from Excellerator (French website, but Google Translate will do a very okay job of it).
Jump ropes are cheap: buy a leather one and a heavier PVC one (see also further below).
(Steel is more violent when you will occasionally whip your back or butt, as I did around a dozen times in my first two weeks. I think it's best if you already have acquired technique and are aiming at a lot of strength gains, but you can get that too from heavy PVC ropes.)
Also of course different lengths, how do I know which length is right for me?
Crossrope's website has good advice about that, but it's a non-issue on most ropes, which are adjustable. I'm ~ 1.75m and the standard 2.85m ropes I bought from general sports stores in my country work perfectly.
Some ropes are called speed ropes, is this just another name or is it really a difference?
It's not just the name, it's really meant to go faster (look at how the rope connects to the handles: the angle is such as to allow for maximum rotation speed).
I have one of those, but as a beginner who's only learnt to do double-unders ten days ago, I believe that this rope will only become fun later. Right now, I'm much more happy using a standard unweighted leather rope, plus a heavy PVC one.
There are weighted grips available and grips that explicitly mention that the grips are not weighted. I am somehow reminded of jogging with ankle weights here, which supposedly isn't useful.
Don't know about jogging with ankle weights (which sounds like torture, I jog and would never do it). However…
… Weighted (heavy) ropes are great. I swim a lot of front crawl, and was looking for a way to sustain my shoulder and arm muscles by jumping rope. Using a heavy PVC rope does that. It changes the amount of effort completely: check point 3 of this infographic.
(Crossrope does what looks like great heavy ropes. Most other brands I know of sell heavy ropes where the weights are in the grips, while Crossrope puts all the weight in the rope itself, with slightly different handles.)
A heavier rope is also slower -- great for learning. I really love it, and have bought weights for my previously-unweighted ropes. Note that using weighted/heavy ropes will require adjusting your workout time.
Does it matter how the rope comes out of the grip: straight or sideways?
See my point on speed ropes (sideways). For beginners, I recommend what you hint at in your next point:
Some grips have ball bearings for the rope, sounds reasonable. To take a break just because the rope is twirled wouldn't be motivating, but is this really a concern?
In my experience, it's not much of a concern. Most of my ropes, however, do come with ball bearings (example).
Last thing I spotted was that there are indoor and outdoor ropes […] should I simply buy one for each purpose? They aren't that expensive after all.
I've been using leather and heavy/thick PVC on rough asphalt, twice daily (25' each time) for over a month, and the ropes are doing fine. See my very first point: leather is really good.