If you begin to activate an ability that costs mana before activating any mana abilities you need to pay for it, that ability will be the one to get Tezzeret's discount. The total cost for that ability becomes locked in, and then you can activate mana abilities to pay for it.

If Tezzeret's second loyalty ability causes a Vehicle to become an artifact creature, it doesn't count as "crewing" that Vehicle for any ability that would trigger off of a Vehicle becoming crewed.

If you activate Tezzeret's second loyalty ability targeting a Vehicle, it will have its printed power and toughness.

Tezzeret, Betrayer of Flesh's first ability reduces the cost of the first activated ability of an artifact that you begin to activate each turn. The second ability (and any subsequent abilities) of an artifact you activate that turn won't cost less to activate, even if the first ability you activated didn't cost any mana to begin with. Notably, this does not exclude mana abilities.