I'm playing the IP against Verena and I have faced her before but no other player has purchased the improvised cover. Consequently I am being annoyed with Her increased survivablity but one assumes that she can use both Duck and weave as well as improvised cover. It doesn't help that she has a combat coat too.

So, can Verena stack duck and weave with improvised cover? I can live with that but please tell me she can't count multiple adjacent figures for using these abilities!

She can use both, but each ability can only be used once per timing instance. So since duck and weave says +1 evade (as opposed to +1 per...) that's all it can give her.

Yup. Only one Duck and Weave, thus only one evade per attack, and only one Improvised Cover per attack.