Since you have several questions I'll number my answers to match the question numbers.
1. Gas checks are mandatory.
2. Because you want to shoot in many guns, get a .226 diameter, which should work in all, and probably a 60 gr weight would be most likely to fit, though it may be too long for some guns.
3. Any of them will be twisted fast enough to stabilize 60 gr bullets, but the concern is more with bullets being too long than with being able to stabillze them.
4. Only way to beat blue soft is with a club, though I've never heard of anyone trying that, of coarse. In fact I'm sure I'm the only person who thinks crazy enough to write such a thing.
5. Rich alloys are best for 22 bullets as the small amount of lead doesn't keep the molds hot very well, so a runny alloy casts best, and hard shoots best.
6. I've only known of a couple customers who talk of making autoloader cycle reliably with cast loads that shoot accurately. It is really difficult to maintain accuracy at the pressures required for consistent cycling, and when shooting rapidly, barrel heat can quickly cause leading of the bore and gas ports. Experiment with the slower suitable BALL powders. Ball powders burn coolest, and slow burners will retain the highest chamber pressure after the bullet passes the gas port.
In case I've not made myself perfectly clear here, cast just simply are not popular in 223's because the majority of rifles in that chambering are autos, or maybe I should say the majority of rounds fired with that cartridge are in autos, which people like to shoot rapid fire. The fun for most shooters is gone when they can't trip off a clip full fast as they can get on target, and that is definately NOT cast bullet territory.