We put in an offer and did an inspection as soon as it was accepted. 

Upon inspection of the apartment that initially reported the mouse issue, the apartment was in the same condition it was a month ago.

But talking with the current owner about the tenant and seeing the unit during our inspection, I don't see any reason not to continue with them.

Im planning to offer with contingency to home inspection, if we move forward.

Like a fire safety inspection or landlord registration or CO inspection?

What specifically do they say is wrong with the windows and doors that cause them to need replaced because this is a large expense and if they are not functional your inspection should show that.  

According to the After the TDS is delivered to the buyer they have 3 days (if delivered in person) or 5 days (if mailed ) to cancel the contract The law does not tie the delivery of the TDS to the inspection.

In terms of the contingencies you can include to prevent you finding something that will evaporate cash flow, use a inspection contingency.

There's podcasts and books out there on home inspecting that can give you a broad idea of what to look for in newer and older homes, but nothing teaches you like experience.

On the other hand, if they are returning your deposit-I'd also ask for interest and any other fees that were paid (eg inspection, appraisal, etc).