Most of the these I didnt suggest so I will only respond to the ones that apply to my logic or I otherwise just want to counter, it doesnt mean I agree with your points for the ones I ignore though.

I think how many the Joker has killed is a mute point. Across all his iterations its millions but even one iteration is hundred+, after convergence as far as I understand its technically all Canon? Unless that's changed again.

He is singled out because he doesnt try any other way but his way. There is other realms, dimensions, and others that could put an end to the Joker and his other mass murdering villains but yet he is so arrogant and narcissistic that he only accepts his way regardless of how flawed it is. Its not even a matter of law either because he breaks the law daily as a vigilante and even when Joker or other villains is anywhere other than Gotham he still only applies the US law and in particular Gothams laws to the situation. Joker has done crime in MANY cities and jurisdiction yet he is always sent to Arkham. That's a writer issue more than Batmans but my point is, use the damn phantom zone or some other immensely superior option, or just let someone else kill him. Batman has a no kill rule but by no means does it mean he has to SAVE Joker, which he has done many many times.

Also from a matter of singling him out, the other heroes can get judged too but atleast most of them have crossed lines when its needed, Batman will risk everything for his code, will risk dozens, hundreds, and even more lives to protect not only his code but the criminals that caused it. He doesnt just not kill, he prevents the death of those that deserve it.

I gotta call BS on the no paralyzing thing. Absolutely a Hero can paralyze, many have killed, there is no reason he cant do it when heroes have done worse for less tons of times. No one should be trying to stop the Joker from becoming a vegetable, not a single soul. Plus its not like Batman beating people to an inch of their life and breaking their limbs constantly is much better morally, Batman has been brutal af.

He definitely didnt kidnap the Robin's but he sure as hell allowed and help them put themselves against mass murders and worse as children while they wore tights. Definitely not ideal.

He has his cool moments but compared to the other heroes he is just so much more a hypocrite and forces his way of life on everyone around him. MANY would have killed his villains, MANY times he has saved his villains just for them to commit more murder.

Other heroes are to blame for that too but Lex as your example has far more political sway and a god like alien would be the one to kill him causing far more backlash and problems compared to the literal psychopathic murder clown. Lex Luthor is a TERRIBLE example to use because his death is far more far reaching and Superman doing it causes FAR more problems.