If you have Alix T4LBO7 rezzed and play a shippment from Mirrormorph, then install a bunch of cards from hand, do each of them count towards the power counters on Alix?

Yes

Does Howler count as well? I know Oversight AI does not.

Anytime you install a card. Of course Oversight AI doesn't work, because it merely rezzes a piece of ice that has already been installed.

Howler also triggers EtF for gaining a credit