Given that your average pointy military jet is these days covered with all manner of radar avoiding, low observable paint and coatings - how do they do that?
Only stealth aircraft I've ever seen with special markings was an F-117 with a huge US Flag on the under side. I would suspect that on most normal military aircraft those areas of RAM paint, which are in limited areas like intakes are left original colour, as are radomes and aerial housings. The majority of military aircraft paint is low observable in IR, not RF frequencies, though RAM 'Iron Ball' paint does exist, it needs special application to make it effective.
I'm sure it's a little more sophisticated than the crew chief and the lads sitting down with an outline drawing of the jet, coming up with a design and then heading off down to B&Q to grab a few cans of spray-paint (although I note that some Vietnam-era BUFFs and even a few Op Corporate Harriers were literally hand painted with brushes at times for operational reasons).
It is, somebody designs it, either a bloke on a unit or they get some artist to design it (or a mix of both) and they get a professional painter to paint it, Either service painters or civil contractors or a mix of both. Car paint has been used in the past (though it would normally be a two part epoxy paint). In the case of the UK (RAF definitely and most likely the other armed services), the design has to be approved by CinC Command and the Camouflage Working Group and the unit has to pay for it. More than 25% repaint requires aircraft to be reweighed (and re-ballasted if out of limits on C of G). In a few cases there is a major use of vinyl decals which are printed, applied and then sealed down
Does someone on the Squadron designate a particular jet that is to be resprayed for such display purposes?
Yes, normally the OC, though in the case of the Germans its based when the aircraft is next due for deep maintenance. In the case of the one I was slightly involved with the original jet selected (the boss's one) couldn't be used due to the fact that it was pinged for a deployment and when the design was going through the chain to the CinC, the CinC No. 1 Group asked if an aircraft could be painted in Desert Pink. The aircraft selected for that scheme was the only only left in service which had flown in the first Gulf war.
Is there a special type of paint that goes on over generic "RAM Fast-Jet Grey" and able to be washed off a few weeks later (cos I'm sure they don't strip em back to bare aluminium and composite).
The temporary paint for normal airframes is Alkali Removable Temporary Finnish (ARTF), and it tends to peal off rapidly. In the case of RAM surfaces, it is noticeable that the majority of stealth aircraft are very lacking in any colour other than a few shades of grey. They most likely have special paint and you're limited on the colours. Most high quality paint jobs on a normal aircraft would need some form of airframe prepping to allow the new coat to key to the paint surface already there or it would peal off in a very short time as ARTF does. This prep ranging from a wash, though a light rub down to a complete strip and re-prime. In the case of the majority of airframe specials, the finish is permanent and the aircraft will have a full strip down and repaint on its next depth servicing, one of the reasons that the RAF limit their 'Operational types' specials to just the Tail and Spine.