Is there anything I can do to reduce the sensitivity of the police? I can't count how many times they have been a hindrance instead of an actual game play mechanic that adds value. A mission I was in involved me carrying a body, I got shot up by the cops, unable to drop the body.

The Delamain missions are my most latest example, I'm suppose to damage the car, yet the moment nearby police hear shooting, they immediately start firing at me, making it absurdly tedious to do a lot of the side gigs. Then you have dumbass civilians running up to me while I shoot gang members, and POOF the cops teleports and make everything far harder than it needs to me.

Is there any mod, any console commands, literally anything I can do to turn them off, or at least make them more sensible?

I agree.
This is one of those things that needs a massive overhaul!

I also have another suggestion: use melee weapons to avoid hitting civvies... . I can't remember how many times I've accidentally tagged the 'victim' in a crime-in-progress and the police teleport in and start shooting.

Only time this didn't happen was when they were already in a fight with some gang-bangers... but then once all of those were down, they got pissed at me, despite me helping them!