Electrical – use an adapter to connect a welder to a dryer outlet

The answer is it depends on the appliance. As long as the appliance does not expressly forbid using it on a 30A circuit you should be fine. You'll be able to change the plug and put on a 14-30P plug, just ignore the silver terminal (neutral).

You can run the sub for the barrage to the house sub panel. As far as the garage panel size I would go bigger, put at least a 100/125 amp panel in this way if later you need more power you only need to upsize the breaker in the house feeding the garrage and the wire gauge. Larger panels are not much more and I have never been asked to down size a panel but have upsized many over the years. For the breaker size on the compressor get the specs off the compressor. Compressor motors may not have "true power ratings" they use locked rotor current. Not FLA + for disconnect and wire size.(note your 16.5 amp size would work on a 20 amp circuit but the mfg instructions need to be followed). For example I have an 7 or 8 horse compressor that only requires a 30 amp breaker on 10awg wire but a true 5 horse compressor that draws more. The only way to know for sure is to check the mfg install instructions. At 100' usually the the wire size won't be needed to be upsized for compressors and welders because they are not continuous loads. With that 240v compressor 4 each 120v 20 amp circuits would be fine possibly more in a 60 amp panel but I would install a larger panel just in case of future needs.