I had specified 'mice' for this week's questions, but I should have generalized it to rodents, since there is considerable overlap among the various disease-carrying species.
Tom made a great start in answering the questions.  I will elaborate below.  Keep in mind, this is just a list of the more common diseases - there are many more!
1.  List 5 infectious diseases spread by rodents:Tom says:
1. and 2. Rat bite fever (
Streptobacillus moniliformis, Spirillum minor)
3. Hanta virus
4. Lymphocytic  choriomeningitis
5. Plague & Leptospira
6. heebie jeepies (right Pritish?)
Here are some other diseases:
Viral:
Lassa fever, Argentine hemorrhagic fever, Venezuelan hemorrhagic fever, hemorrhagic fever with renal syndrome, Venezuelan equine encephalitis, and tick-borne encephalitis.
Bacterial:
Rocky mountain spotted fever, Boutonneuse fever, scrub typhus, murine typhus, tick-borne relapsing fever, Lyme disease, and salmonellosis (with 
S. typhimurium)
2.  Rodents can act as reservoirs for which parasitic infections?:Babesia
Toxoplasma gondii (named after the African rodent it was first described in, Ctenodactylus 
gundi)
Chagas disease
Cutaneous and visceral leishmaniasis
Echinococcosis (E. multilocularis and E. vogeli)
Angiostrongyliasis
Schistosomiasis (
with S. japonicum)
Hymenolepsis (Both 
H. nana and 
H. diminuta)
Tom also gave us the following:
"Just remember, these rhyming species carry lots of nasties: rats, bats, & cats (not sure about wombats...)"  Although, 'man's best friend' doesn't get off the hook either - more on this in a future case of the week...
It is important to note that rodents can spread disease in a number of ways:  the infectious organism can be excreted in urine and then be inhaled following aerosolization (e.g. Hanta virus) or ingested (e.g. Leptospirosis), or the organism can be directly inoculated through a bite (e.g. rat bite fever).  Vectors can also be involved such as fleas, ticks, mites, mosquitoes, sand flies, and reduviid bugs.  Finally, rodents may harbor a stage of the disease that is not directly infectious to humans and must first be transmitted to another non-human host (e.g. echinococcosis).