Character improvement certainly is not required for it to be a roleplaying game.
I think it is. It doesn't have to be mechanical improvement. It can be circumstantial to the game world.
Your decision space is always limited in some way, in fact without limits it is not an RPG either, it is just making up a story.
I disagree. Story space is limited, too. In RPGs we just do most of that limiting beforehand. We want to play an RPG (as opposed to any other form of game or entertainment). We want to play X genre. We want to play Y specific RPG that conforms to that genre. We want to play Z Edition of that RPG. We want to play A-style characters. We want to play in B setting. On and on.
I've been in lots of one shots over the years for a variety of things marketed as RPGs. Do (4) and (5) only apply to continuing games?
I would say no, they don't only apply to continuing games. Persistent world meaning it's internally consistent from one moment to the next, not continually played in.
What does "improve" mean - accomplishing goals the character might have, or improving stats? It feels like a game could easily not need the later (at least after some point).
Sure. That could easily be changed to something like "characters changing based on the decisions you made". It doesn't need to be improvement and it doesn't need to be mechanical.
And, for (1) it sure feels like some decisions are required, and that the number of decisions we make IRL is impossible... and that there is a lot of territory in between that folks here would call role-playing.
Sure. Zero decisions precludes it from being an RPG while infinite decisions is utterly impractical.
For me, it's about having decisions that are relevant. Meaningful, consequential decisions. I don't want a DM to pretend I have a choice when I don't. If I don't have a meaningful, consequential choice, don't present me with a choice. Tell me what happens. If that means the game is a long series of the DM narrating to the players what happens and the only choices we really get are at character creation, then it's quite clearly not an RPG. Likewise, it's not really an RPG if the only time the DM stops narrating at the players is when they get to combat. It becomes a miniatures skirmish game, not an RPG.
And none of that is onetruewayism or badwrongfun, it's pointing out that there are categories of games. Games are fun. Yay games. But chess isn't an RPG. Warhammer Fantasy Battles isn't an RPG, but Warhammer Fantasy Role-Play is. Jenga isn't an RPG in and of itself...but it can be the random mechanic for Dread (or some other game that uses the tower for the random element). Certain elements must be present for it to be an RPG, take away enough of them and it's no longer an RPG.
If you point to a car and say truck, you've made a category error. They're both vehicles, granted. But a car isn't a truck, nor is a truck a car...unless you're talking about that abomination the El Camino. The mullet of vehicles.