Well first off, welcome to the forum.
This guide is a good starting point:- https://z4-forum.com/forum/viewtopic.php?f=2&t=21281 But the BMW Warranty link doesn't let you get the VIN from the Reg No. any more.
Your probably right to look for bigger than a 2.0 litre - that way you'll get a straight 6, which is one of the Z4s unique features.
Pre-facelift E85s (03 to early 06) came as a 2.2i (170PS), 2.5i (192PS) or 3.0i (231PS). They all used the same M54 engine the same as the E46 3 Series, just in different sizes so the running costs are pretty much the same for all of them - so you might as well have the bigger ones!
Facelift E85s came as a 2.5i (177PS), 2.5Si (218PS) or 3.0Si (265PS) and use the N52 engine as fitted to early E9* 3 Series, again just in different sizes - so the running costs are pretty much the same. The E86 only came along with the facelift, and was only offered with the 3.0Si engine. The N52 engine is a newer generation, and has a couple of foibles in that there is no dipstick (you check the level using a button on the dash) and has an electric water pump which seems to last 8 to 10 years or 60,000 miles. Because it's electric when it goes it just stops, so in about a mile the car will overheat. Replacements cost over £200 just for the part, so it's worth bearing in mind - if you find evidence of a car having had a new one that's a bonus!
Road tax on all of these is £325 a year. It's £555 a year for an M which only came along with the facelift, but you'll struggle to find an M in your budget unless it's very leggy. And in any case everything for an M costs significantly more - plus they need valve clearances checked and adjusted regularly as they aren't hydraulic. The upside is you do get 343PS though. :laugh:
One issue the E85s suffer from is roof motor problems. BMW located them where they often get drowned in water from the roof drains. I think BMW want in the region of £800 for a new one, but there are people on here who can often revive dead ones and relocate them into the boot out of harm's way for around £200. They are in Hampshire, Bristol, Warrington and Durham, so it depends where you are.
Some E86s have suffered from rust on the tailgate - usually around the edges as you look at it when it's open.
BMW were a bit mean with spec so decide what items are "must haves" before you start looking. Heated seats are very desirable, especially on Roadsters, Bi-Xenon lights are much better than Halogens and M-Sport seats are sought-after - to the extent a pair of used ones will fetch the best part of £1,000. If you stick to facelift models the Sport models had them as standard, whereas they were an option on the SE models.
Other options were multi-function steering wheel, rear parking sensors, folding door mirrors, cup-holders and cruise control, but that last one isn't too hard to retro-fit.
You ought to get a good E85 3.0Si within your budget, or towards the top of it a good E86 3.0Si with maybe 70K or so miles on it as Coupes do seem to command a premium. Higher mileages aren't an issue so long as the car has been maintained properly. I sold an E91 325i earlier this year (same engine as a 2.5Si) that had done 139K miles and was still running fine. My E86 has done 90K and also runs just great. Some have done over 150k miles on these engines without problems.
As to where to buy I'd start on here, as most people on here are obviously enthusiasts so tend to be fastidious about looking after their cars. But I got a good deal on my 1st Coupe on AutoTrader, and it had full BMW history. I sold it on here, but found my next one on Pistonheads.
Personally I prefer buying privately as that way you get to weigh up the seller, and can form an impression of how they are likely to have looked after the car - you don't get that buying from a dealer, and you just know you are paying a premium. OK you get some come back, but quite often it isn't much use!
Anyway that's enough waffle from me - good luck tracking the right one down. :thumbup:
North West Surrey
P) for parts. Also worth considering the M is manual only and has the better hydraulic steering.