I would think that if you use the record option in your DJ software that, yes, you would have stable volume levels and they certainly shouldn’t be influenced by your monitors/master fader (assuming they are master output, not booth).
You can use any tool to EQ/post-produce/prep tracks/mixes for publication, but I’d say that a) Audacity might not be the best tool and b) mastering stuff is a specialty among sound engineers. It’s not a set-and-forget or use-default-settings thing.
There is good mastering software out there, but … expensive.
Without at least the basics of compression for example (5 parameters there!) it will be hard to get things to sound right.
Finally, you are a DJ and as such use prepped music. If you use anything remotely modern, not only will it be professionally mastered and EQ-ed by engineers, but it will be compressed to the hilt already. To then go and compress a mix even further is gonna have you end up with a sound as flat as a pancake and not sounding very nice. Tiring and boring comes first.
So, figure out how to do proper recording in your DJ software, keep an eye on channel gaining (0dB rulez) and only move your channel faders for mixes or for effect. This should give you a clean sounding mix. Don’t compress or EQ anymore. What you CAN use audacity for is to correct mistakes you may have made without having to redo the whole thing.
Clearly the Pro Mixtape course is THE place to go to get really serious about your mixtapes.