Nightwing and Batman visit Joker in Arkham to see whether he was related to the Red Hood in any way, which he is not. However, his actions caused Jason Todd's death and subsequent transformation into Red Hood after his resurrection, so in a way the Joker was linked to Red Hood in more ways than one.
Red Hood seems to use an awful lot of explosives, to the point of it becoming signature. Joker killed him with an explosion, and he probably got so traumatized by it, that it became his signature.
Fridge Tearjerker: Batman's no-kill rule to villains, as seen in this feature, is not by any means a sign of altruism or value of life, but it's the only thing keeping him from becoming as bad as the villains he fights. And sadly, Red Hood crossed that line with breaking that rule, no matter how desperate or successful his goals were.
It is understandable why Jason opposes the idea of executing Dent: Dent was a good guy who was consumed by mental problems, unlike the Joker who is a sick man with nothing redeemable about him. Jason, who is more of an Anti-Villain, would have the decency to remember Dent's good side.
A possible reason Batman hesitated to kill Joker, which he may only have realized at a later point, is that given he's just seen Jason Todd come back to life, what's to say Joker won't?
After Joker finishes beating Todd to within an inch of his life, he turns back as he's leaving and tells him, "Please tell the big man I said... hello." At first, it seems he's referring to Batman. But after it's revealed that he's rigged the place to explode long before Batman can rescue him, it takes on a new meaning altogether. He's not referring to Batman; he's referring to God.
The whole resurrection of Jason Todd doesn't quite seem to gel with the Red Hood's violent but entirely sane mindset, to the point that he comments that Batman might just think he's rabid and insane. But think about it; Ra's Al Ghul himself is panting and stressed out by the effects of the Lazarus Pit, as it likely is reforming and rejuvenating his entire body forcefully. When Jason's corpse gets dumped in, he comes out vomiting the fluid and screaming non-stop in agonizing terror; he not only just came back from seeing the afterlife, but is probably feeling every bone, every organ, every part of his body filled and rejuvenated by the fluid from the inside-out all at once in nonstop pain. He had at least several broken ribs and damaged organs from the Joker beatdown, and the explosion probably pulverized his internals; everything would have to be healed excessively to recover compared to simple aged cells. And all of this on top of the Lazarus Pit's momentary insanity inflicted on resurrections, something thus far untested in this continuity.
