Hi,

I'm not using horse combat much because I'm not really interested in it but I recently gave it a try. Result: striking with the left hand works but striking with the right hand is impossible. When I release the left mouse button, the arm of my character remains stupidly in the air and that's it.

I must say that I haven't tried with different kind of weapons; my character is currently using the mace of Molag Bal.

If the arm gets stuck while trying to do a power attack (it seems to be the case from your description), then it's most certainly a mod related issue. More exactly, you have a mod that edits Races and was not updated to work with the mounted combat.
I have seen this exact issue numerous times.

Now, you might say, "Hey, but I don't have any mods that edit races! Don't be silly." It might not be obvious. For example, a mod that adds new face complexions or warpaints is done through race editing.

Just check your mod list and look for Race altering mods and see if they are up to date. I usually load the mod in the CK and check the Attack Data to see if it has MC events in the Race(s) it edits.