Could "thiet ke" be used as a verb as well?
Yes. "I design a shirt" = "Tôi thiết kế một cái áo" :)
This comes from 設計 which can be used in the same sense as:
Does the audio for "lấy" sound a bit off to anyone else?
I don't agree that this is a good english translation. It's passable but could be improved. E.g. "she has taken my design"