Best Way to Integrate Subwoofers?

Correct, the speaker wire from the amp over to the powered sub should only be seen as the signal. It's not actually driving the sub. As for latency, etc... Get a sealed servo sub or 2 or 4 😀. Those should keep up nicely. Volume should go up and down automatically with preamp volume control like normal. Just look for a sub that has high level inputs. REL and Rythmik should fit the bill.

If your mains are not being driven to their full potential currently with your amp then that's a whole separate issue I would takle first. I don't buy the statement that you should relieve the amp of low freq signals from the mains.
Step 1. Get an amp that can adequately drive your mains
Step 2. Worry about the other stuff

"So - if the subs have their own amplifier, then both the main amp and the sub amp are driving the subs, and you adjust the subs volume down to mate with the volume of the mains, and then the preamp volume control adjusts both volume levels in synch?"

bassdude-
the sub isn't being driven by the main amp. The impedance is preventing that. The signal is just high enough to be amplified by the subs amp.

The REL method is simple enough to try compared to the other suggestions.

This is a very complicated problem. Forgetting the variability of speakers and subs there is a best way to do this but the market has not responded yet by providing the equipment options required to do this correctly. What is needed is a free standing digital bass management processor that would slot into the equipment chain before the DACs. You would need four DAC channels one for each sub channel and one for each satellite channel. The first company to get this right was TACT Audio which was put out of its misery about a decade ago. With their system you had complete control over frequency and order and you could change settings on the fly. You could even use different high pass and low pass filters. You could cross all the way up to 10th order if you wanted to all distortion free in the digital domain. It was killer. There is only one unit that takes over where TACT left of that I know of and that is the Trinnov Amethyst. Perhaps Lyngdorf does this also but I am not sure. The only down side for some would be having to digitize their analog sources. Every other approach is a compromise. I personally do not like putting powered electronics in subwoofers. I prefer separate amps and crossovers which allows you your choice of poisons. Several rules of thumb that may upset a few of you. Don't cross so low. Any good sub can run fine up to 200Hz as long as it is in stereo. Right now I cross at 125Hz 48 db/oct. It is much harder to match up drivers the lower you go. Most systems do not need to cross lower than 100 Hz as long as you can use 24 db/oct or higher filters.  I do this with electrostatic panels by the way.
Alway use a high pass filter on the satellites. If you can see a driver move you are doppler distorting everything else carried by that driver. Even with  analog filters the improvement in satellite performance will far out weight any detriment due to the filter. In an analog system the crossover should go between the preamp and amplifiers. The improvement you get in overall system performance will far out weight any detriment from any decent crossover. I think the JL Labs crossover is as good an analog crossover as you are going to find. Hopefully there will be stand alone digital cross overs or even DACs with bass management built in. Once you work with this in the digital domain I promise you will never look back. Regardless of the crossover there are a few other requirements for the best bass. A minimum of two subwoofers is necessary for point source satellites. I use linear arrays and in this case the number of subs depends on the length of the front wall. I use 4 subs at 4 foot intervals along a 16 foot wall. This then functions as a horizontal linear array matching the power projection of the satelites. Subs have to be real heavy. Mine weight just shy of 250 lb each. You should not feel any vibration when you put your hand on the enclosure. Vibration is distortion. Getting that kind of performance in an enclosure is no easy feat. Look what Magico resorts to! Build your own subs if you can. You can do a much better job than most companies because you don't have to worry about making a profit or shipping costs. There are any number of great subwoofer drivers out there. The hands down best buys are from Dayton. I also have a soft spot for Morel drivers. There is no better material for making subs than MDF you just have to make it thick enough. The last subs I built for a friend used 2 inch MDF in a balanced force design (driver on opposite sides ala KEF and Magico). I am warming up to build cylindrical balanced force subs using two Morel TiCW 1258Ft drivers in each of four cabinets. I just have to figure out what I'm going to get for my wife in exchange.

1) Running speakers full range creates phase issues below the crossover point, Gedees advocates this but I disagree, you want all “subs” to be the same model. 
 
2) Digital/Active crossovers are the way to go, especially if somewhat tech-savvy. The MiniDSP SHD is awesome and can also be used as your pre-amp (high enough quality to not degrade even state-of-the-art systems. 
 
3) Crossover where the speakers roll-off, you take measurements; for instance, if doing 2nd order on the speaker, find the in-room -12dB point, and crossover at 2x whatever frequency that is. However, you don’t want the crossover to be >100Hz, as then you get localization issues.