Hypertrophy training and strength loss...

[Mrrippedzilla]

How often do you recommend varying the progressive overloading of different variables as listed above?

All of these methods are sub optimal for growth compared to lifting heavier so I only recommend them in specific cases, such as if your unable to lift heavy due to injuries or performance/sport specific periodization programmes (strength/tension/conditioning cycles, etc).

What variables could be changed together?
How would it affect progress in terms of strength, size, or endurance?

I usually recommend changing one variable in order to accurately track that your making progress - most people are terrible at keeping training logs as it is and changing multiple variables can easily result in you simply spinning your wheels.
Having said that, several can be changed at once if you know what your doing. Staley's density programme for example focuses on higher volume within the same time limit BUT he also recommends increasing the weight 5% once you've beaten your total workload number by 20% - this is an example of changing volume & weight while still maintaining a volume emphasis.

Changing any training variable can have a significant impact on strength, size and endurance but I'm not going to go into detail about this because it would be overkill.
Suffice to say that if you want to get stronger than you need to train with heavy weight, low volume - focusing on rest or tempo or whatever is a complete waste of time.
If you want to build muscle then its a combination of tension, damage and fatigue (many ways to do this - increasing weight being the simplest).
If you want to build endurance then metab-olic & conditioning work would take precedence with slight changes in programming depending on whether we're talking about endurance alone, speed-endurance or strength-endurance.

Maybe even... what is the method you've seen be most consistent in causing growth and allowing strength to improve simultaneously? Particularly in individuals who are not beginners.

Doing both hypetrophy & strength training simultaneously

Its not that difficult to have elements of both within the same workout, or the same week or in a mesocycle if you actually know what your doing (which I do) and there is no set "method" that is supreme in this regard - all context dependent.