Good advice about the A/C. Either inspect it yourself, or have someone come out to check it.
I bought a house that was a top to bottom remodel. During the remodel, they ran the main floor A/C unit without a filter. So all that drywall, wood floor sanding and other construction dust got into the evap coils and clogged them up. Unfortunately, it passed inspection and was working fine on the day I bought it.
But literally the next day after I bought it, the compressor started short cycling, probably tripping on high head pressure due to the evap coils being clogged with major shit and iced over. This was a monster 6 ton all-in-one unit with A/C and gas heat that was outside and probably cost north of $10K. It would have sucked if it took a dump because of negligence, especially since it was only a few years old. In any case, it might be a good idea to get an HVAC guy to inspect the systems and don't rely on a home inspector.
In regards to basements, make sure there are no water leaks and it's dry. If there is a sump pump, make sure there is a battery backup.
One thing about basements is it's very easy to accumulate a lot of crap down there, since it's a big storage area. Try not to fall into the trap of storing too much useless crap because if you ever have to move again, it will take months to clean it out. Also, always run a dehumidifier in a basement.
Like Frank said, get a pre-approval letter if you're doing a mortgage. In this market where cash is king, you'll need one to even have a chance.