You could even go to a 60 amp breaker and be fine with the #6 wire feeding the receptacle, if you needed to. Depends on where you look it up, you will see 50 and 60 amp being the commonly stated continuous duty ampacity for #6 wire. To really cover all bases, an elect. contractor or electrician would look at the temp ratings of the devices and the wire insulation type, and look up the ampacity for #6 copper wire for the ambient temp in the area where the wire is located, but if you bought a modern receptacle and use THHN or THWN wire or cable like you probably did, you are all set unless this is located in a hot location where the ampacity would need to be derated (larger wire would be required). The beauty of making it a standard receptacle instead of derating the wire size for a dedicated welder recep, is that it is appropriate for other things as well. A kiln is a common high amp continuous duty load that would not be proper to power from a derated welder receptacle. It sounded like in your original description that the distance from your panel to the recep is short (?), so it would be easy if you ever decided to up the breaker and wire size if you started tripping the breaker. If you wire it for continuous load, you won't have to worry about it being a special purpose recep. Just keep the breaker size appropriate for the wire size.
It does not matter in what order the smaller and larger wire are in, as far as voltage drop with your ext. cord. If the machine came from Miller with a #8 cord, then as long as you stick with that size or larger on the machine, it doesn't matter if you decide to install a larger size cord on the machine. Just don't go smaller. My welder came with a 6' long cord which irritated me being short, because it meant I would nearly always be using an extension cord, so I opened the case very carefully, and replaced it with a 20-something foot cord and a 6-50 plug. And I bought cord with a temp rating on the isulation around the conductors that was higher than the original cord had. Hardly ever will need an ext. cord now. You might want to consider that, especially since your box is already pretty heavy....a few more # of cord won't hurt. Also make sure if you buy cord for the machine or to make an ext. cord, you get stranded wire, not solid. It needs flexibility.
Sure, you could put a 6-50P plug on your compressor and run it off that welder receptacle, as long as that source meets what the mfr of the compressor specifies on the nameplate. I assume you are talking a compressor that is also 240 volt, 1-phase AC input like your welder? Then look for what the full load amps needed is. Check the manual for the compressor, if you have it. I can't imagine it needing more than 50 amps unless it is a bigun. BUT....sometimes a piece of equipment will also specify a MAX limit on amps the source can supply....I think that has to do with limiting the amount of current that would be supplied in case of a fault....but the electrical pros can better comment on that. If your compressor also needs 240v, single-phase AC, and the full load amps is under 50, I would plug it in and go. Ideally the most load you should intentionally place on a circuit continuously would be 80% of the circuit's ampacity..... but a compressor may also have some allowance built in to the name plate rating for it being intermittent. I don't know if they are considered continuous loads.
Electrical pros?