We've seen autonomous departments within the Federation recruit experienced officers before, such as when Section 31 added Dr. Bashir to their ranks, albeit against his wishes. The following question must therefore be asked: would the DTI have any interest in recruiting Spock? Or, perhaps a better question might be, has Spock already worked for them, thus explaining his proficiency in temporal mechanics in the first place?
Well, Section 31 isn't really a "department," it's more of a secret cabal within Starfleet. And that's the key -- its members are Starfleet personnel. The DTI is a branch of the civilian government of the Federation. If you look at the United States FBI or ATF or DEA, you'll find that its personnel are civilians, not officers of the armed forces. Some of them may well be former military personnel who've returned to civilian life, of course, but they are civilians.
Certainly the DTI has worked in cooperation with Starfleet personnel on various occasions, since Starfleet is the branch of the Federation that most often gets involved with temporal phenomena (which is why Spock has so much experience with same). But they're separate institutions. Starfleet officers often work with
DTI agents, but military personnel do not work for
I'm sure the DTI would've been glad to have Spock onboard as a consultant or investigator if he'd ever chosen to retire from Starfleet, but the only time he ever did that was when he returned to Vulcan to pursue Kolinahr
. So if they made the invitation, as they most likely did, he clearly never accepted.
Section 31 prides itself on covering its tracks, but the DTI can completely erase their tracks from existence if they so desire.
No, they cannot. At least, they would not desire to. That's misunderstanding their purpose and resources. The DTI investigates time travel and attempts to limit it, contain it, and deal with, or at least identify, its consequences. It doesn't actually engage in time travel except in the direst emergencies.