Depending what the price is, I'd probalby look for one that has been slightly less modified/beat. If it is cheap enough, and you have the place/money/time to do some maintenance on it, it might be an okay buy though. They are fairly easy to work on and parts are readily available.
That one obviously has aftermarket exhuast, aftermarket I/C pipes, aftermarket BOV, open filter element (probably K&N), a leaking oil filler cap (which is extremely common with turbo DSMs), and a leaking valve cover. If the boost has been turned up at all, which I'm sure it has based on the visible mods, the clutch/flywheel needing replaced at 65K wouldn't concern me.
With this car having 65K on it, you are going to HAVE to replace the timing belt, water pump, oil pump, front cover, balance shaft belt (or remove it and the balance shafts) and all the tensionsers and pulleys - if you don't have 100% proof that it has been done. If you don't do the work yourself, this is probably a $500 - $1000 repair, depending on brand of parts and labor cost where you live. They are interference motors so you don't want to push the timing belt life - especially with one that probably has had oil leaking on it.
In answer to your original question, having different tires itself isn't a huge deal - having different diameter tires is a big no-no for the AWD mitsus. So if one is brand new and the other three are pretty worn down, those other three are going to need replaced too.
Dan.
-edited to add-
Considering it sounds like you want to save money due to the purchase of a house, I would personally pass on this one.