Is it ok / legal / allowed to copy some of the styles from stackoverflow / serverfault / etc. for use in a stack app?
Have a look to Trademark guidance.
Design of Your Product
- Do not copy the look and feel of our products.
and quoting Joel answering a previous question:
Just be reasonable. Trademark law is based on the principle of not confusing "the reasonable person" as to the origin of goods and services.
So, i don't think it is a good idea to copy StackOverflow's CSS for your apps.