Okay, I got this Ask from @frostbite833 too long ago, and forgot to answer it (I'm sorry!) then accidentally deleted the actual Ask. Luckily, I'd copied it so I could type out my answer off tumblr.
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If Jason Todd had to take over as Batman for a year or two while Bruce Wayne was under investigation for some major crimes he didn't do and er, Ric Grayson was unavailable, how well would Todd do as the Dark Knight in fighting crime, saving lives and cooking up plans...without killing anyone? Also, how much time would Gordon and the rest of the GCPD put in on proving Wayne's innocence? Note: This "replacement Batman" scenario could happen in ANY DC universe.
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I think Jason probably could do a good job as Batman, but I don't see him taking up the mantle at all, honestly. I think it would fall to Tim. (And how dare you even mention Ric "Edgelord" Grayson in my presence.)
Jason is smart. He is a planner, and though he probably wouldn't take it as far as Bruce does, I don't see him having any issues on that front.
Jason is tough. As a child, he was trained on the streets, and then by Bruce as Robin. After he came back, he was trained by the League of Assassins, and it's been speculated that he's the best, or at least top 3, fighters in the batfam. His biggest problem is his temper/emotions. So, can he fight crime, and do it well? Absolutely, no question. Does he want to fight crime handicapped by Batman's rules? I don't think so, especially with no Batman to answer to.
As to the last question, I think the GCPD is going to do their best for Bruce. I mean, it's been heavily hinted on occasion that Gordan knows who Batman is. Plus, you've got Barbara, Kate, the batboys, etc. presumably working the case. Not to mention Lois Lane, the best investigative journalist in the DC universe.
So anyway.. that was a lot. TL;DR Jason would absolutely make a kick ass Batman if he wanted to, but he wouldn't want to be handicapped by the rules.
I don't know if @crimson-cowl wants to add or correct anything as I'm by no means an expert on all things Jason.