That price sounds like an okay deal, not an awesome one. I got my ’05 with 140k for just over $5300, though mine is a bit older and it’s a different motor. Make sure you get the adaptive xenons, they’re easily the best thing about modern BMWs.
2007 had the N52 motor, and the cooling system has different problems than the M54 in the pre-07s. The N52 has the electric water pump that has been known to fail suddenly, but by now, it should have failed and should be on its second pump and the revised version tends to be pretty reliable. The N52 likes to leak oil, look for oil filter housing gasket leaks, oil pan gasket leaks and valve cover gasket leaks. The early versions of the N52 had issues with broken “external headbolts”. It’s my understanding that these didn’t really do much to retain cylinder pressure, but they can cause a (you guessed it) oil leak. I had an ’06, ’09 and ’11 N52 and I didn’t have any issues with that, but the internet says its common. All three N52s had the oil filter gasket leak though, and my ’06 had a valve cover gasket leak.
The rear suspension design is the same as the E46s, so look out for the same stuff you would on an E46--driveline clunks (Diff bushings), squirrely feeling over bumps (RTABs), oddly worn rear tires. Also, the bottom coil of the rear springs is known to break, so check for this on your prospective E83. Mine is currently broken…I’m putting Eibachs on since they’re about the same cost as new OEM springs. The front suspension is like an E39/E6x/E9x, so look for worn “tension strut” bushings, it’s the front lower control arm. When they’re worn, you’ll get a warped rotor feeling on braking and sloppy steering.
Look to make sure the transfer case fluid has been changed. Worn fluid will cause a really low frequency growling sound at highway speeds that will turn into a shudder during acceleration. I fixed the shudder on my E91 (similar transfer case) with a fluid change, but I’ve heard that it can progress so far that you need a rebuild. Transfer case rebuilds are really expensive. You can troubleshoot by unplugging the transfer case module under the rear trunk floor. The transfer case shudder feels just like a bad coil or a misfire, this will help you not buy all new plugs and coils unnecessarily (like I did on my E91).
The actuator gear on the transfer case also has been known to fail a lot, though it’s a pretty cheap part (~$20). I haven’t done it, but it looks like a pain in the ass to get to. You drop the transfer case bracket, supporting the trans with a jack, unbolt the actuator motor, and replace the gear.
The interior has all of the E46 common issues, window regulators, seat occupancy mat sensors, heated seats that burn a hole in the seat leather. The E83 had probably the nastiest interior of any modern BMW, but I really don’t mind mine THAT much. I’m not much of an interior guy though.