Could Dumbledore and the rest of the Hogwarts Professors pick up another Wizard's wand and identify its owner without having seen the wand before?
Ollivander was able to identify the owner of a given wand by touching it, but of course he had seen the wand before and presumably the person it was purchased for ("The wand chooses the wizard" -- I don't think the wand chooses the wizard by long distance). Perhaps Gregorovitch was able to do this as well. As it is, I personally cannot think of a canon example where someone identified the owner of a wand solely through touch, without knowing the owner previously.
In the Shrieking Shack in Prisoner of Azkaban Lupin identified Harry's, Ron's, and Hermione's wands:
He separated Harry's Ron's and Hermione's wands and threw each back to its owner; Harry caught his, stunned.
Identifying Harry's might have been simple because when he originally disarmed them, Harry's was separate. However, Ron's and Hermione's were both disarmed from Hermione:
"Expelliarmus!" Lupin shouted.
Harry's wand flew once more out of his hand; so did the two Hermione was holding. Lupin caught them all deftly, then moved into the room, staring at Black, who still had Crookshanks lying protectively across his chest.
This could thus be an example of a wizard being able to identify a wand's owner.
It is, of course. possible that Lupin simply carefully observed his students and memorized their wands, but there doesn't seem to be any reason why a teacher would do that, especially when they have several hundred students.)
There's no canon evidence I know of for the teachers to do so, but the Ministry can identify a spell caster in some cases - such as casting Prior Spell on the wand (assuming they know what spells the wizard did last).
Also, Arthur Weasley in his job went after wizards who bewitched Muggle things - meaning they likely had SOME way of identifying just who bewitched the things.
Probably no. The evidence is in Harry Potter and the Order of the Phoenix chapter 7. When Harry enters the building of the Ministry of Magic, a security wizard records identifying information of his wand. Chapter 34 then confirms that this is standard procedure for all visitors of the Ministry. If it were easy to identify the wizard from a wand without recording this information, the Ministry would probably not bother with that step.