Drivers________ speed laws only when they think the police are near.
a. obey b. observe
You are correct. Either would be perfectly ok there. 
No one needs to feel awkward in _______his own customs.
a. following b. pursuing
This one is more tricky, IMO. I think A would be used by native speakers. I think the distinction is that customs are created in the past, so we follow them. One can pursue a dream (or one's study as you said) because it is something that is ongoing or in the future. 
I hope I make some sense here.
Red5
Webmaster, UsingEnglish.com