From the POV of Dumbledore, there are several unexplained things about Quirrel's behavior

Most of those can be explained away with reference to the others and the poor information available to Dumbledore. What stops Quirrel from saying he blasted off when he detected Hermione had been trapped on the terrace and the troll was immune to sunlight? There are no atomic clocks here. Similarly, if he had some surveillance system, he can say he stopped when Hermione died and there was no more need to hurry (alas); between the wards and the soulsplosion, there's plenty of ways for someone like Quirrel to plausibly get the info without positing bizarre unique psychic bonds.

Dumbledore would know everything the wards reported. If they alerted Quirrell to the location of the troll, why not Dumbledore (or McGonnagal)?

There are multiple levels to the wards, and we don't know what Dumbledore or McGonagal do or do not know - I've pointed out that Dumbledore wasn't even in the castle, it seems, and McGonagal has no access to phoenixes and is not a battle mage who can burn through ancient magical walls at the speed of flight.

Then why didn't he find the troll before Harry did? (See above)

Perhaps he was bored, or piqued at the distrust.

Moderately plausible. But: spy devices which are better than the Hogwarts wards? Or, which allow him to remotely look at any area instead of merely being alerted when a condition is triggered? I don't think Dumbledore approve of him taking such power. And Dumbledore would of course check his story - so the spy devices would need to actually exist, and not have been detected.

We know he admires and imitates Muggle technology - recall the very first class. A spy network is perfectly reasonable. Of course Dumbledore would disapprove and would check for its existence, but sacrificing a spy network seems like a small price to not be the culprit for murdering a little girl. He can always replace it or put in a different system.

Not foreshadowed, hence bad storytelling.

I'm afraid out-of-universe reasoning is not really available to the characters. :)

Then why hadn't he consulted it right away before the professors left to hunt the troll, and see Hermione was out there?

He can claim she was nowhere near and it was not an emergency until she was trapped on the terrace and it became clear that something had gone terribly wrong.

Also, what plausible excuse could he present for having the Map while the Twins are Obliviated? Dumbledore is the one main character who definitely knows of the Map, so Quirrel would want to hide it from him.

For all we know, Dumbledore might have a hand in the Obliviation. And did Quirrel expect the Obliviation to be detected? One suspects not: he wasn't expecting Harry to be there either.

Then, against, how did he know when "The crisis is over" and he could stop hurrying towards Harry?

Already explained.

With future intel, why hadn't he prevented the disaster entirely? Why hadn't he warned anyone?

"I could explain, but I fear the necessary graphs would make your head explode and neither Snape nor Dumbledore could follow my exact reasoning. As Mr Potter might say, 'do not mess with time'."

Possible (although can paintings or ghosts move quickly enough to deliver the message in time?) but again, why did he stop?

Him stopping is really not as hard as you think it is.

Again, consider the alternatives here. That he has a unique, unprecedented, unnamed, inexplicable, unknown-to-all-but-him-and-Potter (AFAIK) telepathic bond and also in conjunction with an elaborate theory about him trying to kill Hermione (why?) this explains some movements during a chaotic 2 minutes or less of troll-hunting based on finely-parsed counterfactuals of what he would and would not know where you suppose that characters would place high confidence in their deductions about an infamously devious, whimsical, highly-knowledgeable intelligent weirdo.