Hello and
Spend a couple of hours reading the forum and the Buyers guide.
As the youngest of these cars are now approaching 12-years old things will begin to fail, and some of those parts are very expensive - the X5 is a premium model and comes with those kind of costs.
There is nothing you would not expect to find on a car of that age, or older. A good one is now harder to find so take your time. When you find one you will love it. A bad one will break the bank!
The biggest common items are auto gearboxes @ £1,600+ to repair (don't even think about new!). Transmissions can be very sensitive to tyre choice and wear depth differences front/rear. This can (but doesn't always) result in diff failure. V8s need checking for timing chain guide noise. Any rattle in that area needs replacement or a major engine rebuild will follow.
The 30d is the most economical returning low 30s on motorways and mid/upper 20s elsewhere. Decent tyres are around £600 a set. Servicing is cheaper via an independent. You can DIY, especially if supported by a Forum like this one. Tax after early 2006 is £500, cheaper if registered before March of that year. I think it was Band K.
You can run winter wheel sets to make the handling even better in snow, and they do drive very well on any set.
Those of us who have had E53s liked them (even with niggles) and would have another. Many of us have gone on to have later versions and we like them too. Get as newer model as you can afford would be my advice.
Going in with your eyes open is the best way.
Never anthropomorphise computers. They hate that.