Is it unprofessional or a bad idea to use my department mailing address for personal mail?
Check with the department/school first.  

In general for use of any work address:
For something like receiving a single package, it should be fine. 
For something like receiving regular mail on a daily basis, no you should not do this.  Besides the issues of policy, there is an increased labor of the recipient to deliver your personal mail.  There may be delays.  And, normally these boxes are in a partially-public area and subject to theft.  

I've been in situations like this, so I got a PO box once and a private mailbox company another time.