Citizen wrote: does this necessitate bringing all the faders down, and restarting building the mix from scratch?

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Secondly, is there a recommended position to have your individual channel faders at in your default template that will generally avoid the overall mix running too hot into the master?

While you can bring all tracks down by selecting them all and move, you may have other issues that make it necessary to start over.
RMS Levels
When I was in your position about 10 months ago I read up on gain staging, experimented and rather slowly made the decision to use -18dBFS RMS levels with the faders set at -6dBFS as a starting point when setting up initial track levels. This meant I had to get a decent RMS-capable meter and I've been using
Sonalksis FreeG.
I feel that -18dBFS gives me enough headroom for peaks before I control them with Transient plug-ins and Compression/Limiting. I set the levels by controlling the volume at the source or as close to the source as possible,
Compensating lower volume
I overcame low monitoring levels by increasing the volume in my audio interface and monitors with the master at 0, quickly realized the perils of that with stray high levels biting into my ears when using head phones, so I use a Limiter on my master that is always on when I work with head phones.
Compression and Limiting
I also started using gentle multi-stage compression. These days I set levels with compression active on busses and a limiter active on the master as lower sounds doesn't have to be so high on the track fader with that on the master anyway. I do turn these dynamic effects off and on while mixing to compare operations, especially when soloing.
Good sound selection, programming of sound sources as well as Transients, Compression and EQ is what make sounds cut trough the mix for me. And if I need to raise the level of a particular instrument if so just for a specific section I have the headroom now to do that with fader rides.
Fixed Levels
Another critical decision I took was to use very fixed levels to work with, by deciding on a loud level, use a dB meter to measure up that level and mark up positions on my equipment. From the loud level I have chosen a very quiet level as well and another in between.
I switch between these 3 and never work without these levels unless testing the material on a specific system that I can't control fully, like a PA. In other studios I set the same 3 fixed levels. This has made it more easy to overcome monitor issues concerning different systems as well as not tiring my ears.