No problem, if it is 5V output you can connect it directly to Vin, instead of power connector jack.
You are absolutely wrong that Vin MUST to be more than 7V, nothing of the kind.What happened if you apply voltage less than 6V?Nothing, read how voltage regulater works, chip will just supply "at most" to the load,in this circumstances Atmega328 will get 5 - 1 = 4, that is good.
Yes, I checked it out.OPA will turn down power from USB, if Vin more than 6.6VAnd what ?This switch works ONLY on USB power line.It has nothing to do with what we are discussing.Vin has no "interrupter switch" , if board is not connected to USB (that is the case) this circuitry (OPA) is useless when you power board from Vin, it is comparator if you know what does it mean,and in my opinion you have know clue.
O'K, let have a piece.Power can go down as low as 3.7V, from formula: Vin(3.7) - dropout voltage(1.0) = 2.7 minimal voltage for 328p chip.There is a disadvantage in powering low voltage, as on board voltage regulate would not filter a noise from power line.It make sense to power low voltage (3.7 - 6.6) if the source is clean, like battery or supply with voltage regulator build in, most Switching Power Unit would be o'key.
is the solar panel charger output(5.0v,0.5A) enough to charge the 14v battery when in sleep mode?