Question in title. If Imp player gets threat as a result of mission guidelines or rebel ally, can he immediately spend it at mission start?

You can only deploy at the start of the mission if the mission offers an optional deployment during special setup, or just after the rebels bring in an ally, but if those are the case, then yes, you can spend the threat to deploy figures.

But you said spend threat, do you mean spending threat for abilities other than deployment?

Nope. Your response answered my question. Thanks!

The threat (counter) starts from 0 at the beginning of a mission.

If Special Setup lets you increase Threat (usually twice the threal level), then it also allows to perform optional deployment.

If rebel heroes take an ally, you receive the corresponding amount of threat and can perform an optional deployment.

Note that threat and threat level are different things. (Threat level can be found from the mission slot you are playing from the campaign log.)