Can two students press charges due to rodents (mice) due to damage to belongings. I.e. Clothes and furniture (bed, dresser, shelf, etc.)?

I'm not a lawyer, but I work in regulatory agency that oversees pest control. You might seek the advice of, or file a complaint with, your local health department regarding the landlord's legal responsibility to provide safe and healthy housing. If your landlord is found in violation of the local health codes, then you might have a case. The fact that you are students is not as important as your being tenants.

If the situation is egregious enough, the health dept may require correction of the problem. I suspect if you are seeking damages, you are on your own, but with documentation of a violation, you have a better chance to prevail.

Yes they should

This is not criminal, so there is no charges to press,
Laws on damage of property vary by location, so you will have to say where you are at.

You can file a complaint with the school, the local department of health,