Everything useful is stored (once again) so this is mostly from memory. Dragoons in the ECW varied enormously in quality, some of the better units being used as a 'fire brigade', particularly on campaign - raiding supply columns, beating up encampments, storming parties etc. Others were at best mounted infantry, with the disadvantage of having no pike elements for obvious reasons, mounted usually on fairly rubbish horses.
There were lots of independent troops of dragoons around, of varying quality. It wasn't unusual for a troop of dragoons to be attached to a regiment of horse, to give some fire support, probably in the dismounted role. Full regiments appear to be rare entities, so the 'weak' wargames formation may well be accurate. (the ECW definition of a regiment of course being fairly flexible). The New Model Army had one regiment of dragoons (Okey's) to 10 of horse, so make of that what you will.
I can't recall a description of firelock equipped dragoons, especially as firelocks were thin on the ground anyway, but that doesn't mean that they didn't exist. They would be rare though. The better equipped ones probably carried the 'Snaphaunce', a precursor of the flintlock, but most probably had the matchlock.
There are few if any surviving dragoon standards, so it's not entirely clear whether they carried square or swallow tailed flags, although the latter are traditionally what they were thought to have carried.
FWIW I've got a few troops of dragoons in my armies, who normally get attached to Horse units in the role described above. I wouldn't expect them to stand up to Horse in a cavalry fight, mounted or dismounted, and their relatively small numbers mean they lack staying power against Foote.
As with most things ECW, there are few hard and fast rules, and the rivet counters are unlikely to be able to catch you out on dragoons!
As an aside, I've just come upon this on the BCW site, a reproduction of the series of ECW articles from Military Modelling in the 1970's which always provided inspiration.
http://wiki.bcw-project.org/_media/note ... rticle.pdf Since I'm struggling to find the articles in the crypt that's a real bonus! Pg 28-29 covers the dragoons.