What is an Enginseer's starting weapons and armor? Is it just standard kit for the regiment? Every other class starts with some sort of weapon. Also, if I take a Ballistic Mechadendrite to start, what pistol can I have equipped with it? My choice or do I have to make a logistics roll?

I'd say a basic Ballistic Mechandrite comes with the generic laspistol as a default (like it did in the other game lines). If you want something else, you need to acquire and probably modify the weapon you want, unless you specifically acquired an already suitably modified weapon, which will be somewhat more difficult.

Do they get starting weapons besides the Standard Regiment Kit?

No. Enginseers are essentially intended to be non-combatants. So just SRK, really.

No. Enginseers are essentially intended to be non-combatants. So just SRK, really.

This.