Multiple breakers controlling the same lights - how can this be?
Call the power company and tell them you lost a phase

This will explain all your symptoms.  I almost posted this answer before hearing your reply, but your reply confirms it.  

The power company should come out and do this for free on a Sunday - they did for me.  

This is what is happening.  Your supply wire to pole L1 (or the other, but let's say L1) is broken.  The 240V loads are connected between L1 and L2.  Some 120V loads are connected between L1 and neutral.  

When a 240V appliance (pump, water heater, A/C, range, dryer) cycles on, it effectively connects L1 to L2 through its low resistance motor or heater.  This energizes L1 at L2’s voltage.   

This in turn allows 120V appliances wired between L1 and neutral to power up.  This current flow also allows the 240V load to function somewhat; the large load(s) and the 120V load(s) on L1 are in series, running on 120V total.  

This is far from ideal, but it's enough that for instance water heaters will warm enough to cycle off again, and the pump may do the same (though I certainly would not push my luck with a pump in a most inconvenient location).  Hence the lights cycling on and off "randomly".    When all 240V loads are cycled off, the whole 120V leg will be dead.