-Can you start right away with the push cage or should you introduce the queen cage 1st with the cork?

I'll paste this one up top, as there may be a disconnect here - my understanding has always been that a push-in queen cage should never keep workers out, it should only keep the queen in, so normal queen introduction protocols still apply, as new queens can be balled in a push-in-cage, of course.  Push-in queen SEGREGATION cages want to be "5-mesh" (5 stainless wires per inch, with 4.1 - 4.2 mm openings). And you want stainless steel. 310 stainless is very nice, 304 is also OK. 

If one is using 6-mesh (the mesh from a wooden bee package), this is something I've never done, as it excludes the workers, and merely creates a larger on-frame Queen INTRODUCTION Cage. 

- Do's and Don'ts of push cages?

The authoritative guide is, of course "A Practical Handbook Of Bee Culture with Some Observations upon the Segregation of the Queen", S. Holmes, 1905.  But copies of the original are very hard to find, and there are several claimed "abridged versions" using the same title and listing additional authors, which do not include the crucial "segregation of the queen" chapters.  This is disappointing for what is surely the best-known beekeeping book title ever.

-Preferred cage size?

I cut and soldered up cages that were 6 x 6 x 1, as I use medium frames for everything.  The goal is to leave enough space for brood sufficient for pheromone purposes, but "minimal" for population control purposes and/or a greatly reduced brood area for mite reduction.

-how long do I leave her in there?

That depends on one's objective.  My goal was to reduce the population of my hives during the (at the time) very sharp death that followed the basswood bloom in Virginia.  If I did not take a hive up to higher elevation clearcuts and burns to work sourwood, the queen was caged just as the basswood bloom started to fade.  The cages remained in place until early August, when they were removed to allow a sufficient forager population for the fall blooms.  Clearly, this was very twitchy timing, and required the tracking of growing degree days and repeated bloom checks to predict bloom dates, a technique that has been rendered almost useless by global climate change and the resulting "random thrashing" of what had been "predictable" weather.   If you cage a queen, you are playing chicken with the next bloom.  If you cage too long, and the next bloom is feeble, you end up feeding to get the population back up, wiping out any gains you might have realized.

-Any other push cage best practices or advice?

They WERE a great way to invoke a near-complete brood break when we needed them to make miticides work, and they WERE a good way to cut down on the hatching of workers who would only consume resources during a dearth, but the risk of being wrong in using them really only for the folks who like things like dual-downdraft Webber carbs, spraying oil on their K&N air filters, and other "performance tweaks".    I would not suggest that anyone do these days what I did to merely try to increase a harvestable crop, and avoid feeding.   

I was swayed early on, as I had read of the very famous book cited above years before I ever gave any thought to keeping bees myself, so "segregation of the queen" was something I thought was a common technique when it is actually a "lunatic-fringe technique". I was young, foolish, and predisposed to think that the science was of more value than the art/craft, when the reverse is true.

I would not attempt any of this these days.  I get too many "brood breaks" when queens go kaput on their own.  If someone asked me, I would classify it these days as a "stupid beekeeper trick" as the rewards are low, and the cost of failure high.

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