When I connect a generator my trailer using the shore power cord, do I need to flip the various breakers on my control pane "on" so that the built-in converter will use the generator power for my 12V devices and charge the batteries? Or does it work with all the breakers "off"?

Assuming you don't have a battery isolation switch, plugging into your generator is just like plugging into shore power. With the panel switches off the power generated will go toward charging your battery(s). With some of the circuit breakers on you will power both the trailer fixtures such as lights and a portion of the power will go toward the battery. In the later case, the effectiveness of the charging will depend on how much power is being used by the other fixtures.

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2007 Toyota Tundra 5.7L
2015 Shadow Cruiser 225rbs

Thanks very much N&K

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2013 Shadow Cruiser 185FBR
2013 F150