How long before asking for a formal offer letter? I have recently been offered a position by a university I had applied to.
After I accepted the offer (through email), they said they would send me a formal offer letter along with other paperwork "ASAP". 
It has been over a week and I haven't received anything.

I'm not sure how long these things should take (nor if they normally send the letter via mail or email). Would it sound rude or desperate to send them an email to inquire about why I still haven't received anything from them at this point?

(sounding desperate my hurt my position in subsequent negotiations, but I would very much like to have some closure on my job search as soon as possible)
I have recently been [informally] offered a position by a university [and] they said they would send me a formal offer letter along with other paperwork &quot;ASAP&quot;. It has been over a week and I haven't received anything.
I'm not sure how long these things should take...

Far longer than a week! The sheer number of personnel involved---e.g., your future manager, the committee that selected you, the head of school, perhaps someone above them, HR, other administrators,...---means the process is slow. Couple that with numerous inefficiencies and you'll quickly appreciate that 3-4 weeks isn't unexpected (and it might even be twice that).

Would it sound rude or desperate to send them an email to inquire
about why I still haven't received anything from them at this point?

It isn't appropriate at this point. Maybe after 2-2.5 or maybe even 3 weeks.

I would very much like to have some closure on my job search as soon as possible

You already have informal acceptance. I wouldn't expect rescinding acceptance to be particular common. (You could always ask a question about that.)