Don't forget to clean it off spit with distilled water or isopropanol afterwards though, as it contains quite a lot of bacteria, which may dine out on the varnish in the long term. The bacterial content of human saliva is the main reason I would suggest that you never blow in to a watch, or breath on to a camera lens to "clean" it.
Slightly more aggressive is white vinegar, which works on certain stains and marks, but needs careful testing with a tiny amount on an area that is not going to be seen. It may dissolve some types of lacquer. Denatured alcohol works on some stains, but again it also dissolves some finishes, likewise acetone, isopropanol, and other organic solvents.
As has been stated, any action runs the risk of further damage. Even tap water, depending on the mineral content, may leave annoyingly obvious marks when it dries.
Less is often more, so work on small areas at a time, use good magnification so you can see the results of your actions close up, and if you get the slightest hint that the varnish is lifting, or the finish is being damaged, stop before you ruin the whole thing.
Yes it should be a tapered screw, it forces the hammer arms apart or lets them come closer together. They are often really tight. Put some oil around the perimeter of the head and coax it in both directions. A little turning usually makes a big adjustment, careful!
You may not need to adjust it though, do you have clearance with the minute cam? Also, the jumper spring that brings the hammer in contact with the cams doesn't generally have a lot of power. It needs to be greased and reliably coming into contact with the cams. The hammer faces will need a light coat of grease where they contact the heart cams.
With a ring inside the crystal this is an armed tension type. The Sternnkreuz 'G22' catalogue covers these which you can download at http://www.sternkreuz.de/accordion/assets/Sternkreuz_Gesamtkatalog G22.pdf
Whilst for the Sternkreuz 'ST'' range Roamer does have some specific citations (p25), for the 'ATC' range you need none are mentioned. Almost certainly therefore you'll be fine with a standard ATC crystal that meets your measurements.
The usual advice is to go 0.1 larger than the outside diameter measurement you've taken. This means when you release the crystal, after it has been gently 'squeezed' in the claw removal/fitting tool, it makes a nice tight fit.
Since you're using Cousins your best bet is going to therefore be https://www.cousinsuk.com/product/standard-chrome-ringed-glass-atc# and the 30.7mm size.
Is not that difficult really. As Matabog noted, dies threaded M6 are standard. The nice expensive set have their Dia. specified, the Chinese write Rolex just to indicate quality, but we know that true Rolex-specific dies are for the screw-on case-backs ...that's another story.
Then on AliX search "watchmaker dies", 20 pieces nylon are like$8, aluminium quite a bit more. I personally choose based on shipping times, things have improved but China post is still very slow, HK, NL post are OK, 7-10 days.