Will this Solar-Electric hybrid circuit work?
Can the battery be used while being charged by the solar panel?  


Yes, while some charging circuits that combine the batt. charger &amp; power supply into one unit don't work to charge &amp; use at the same time (but many also do), using a separate charger &amp; power supply always does.

Would the charger circuit use up the juice when it's not charging the battery?  


The charger circuit will use up some power, even when not running. How much depends on the charger you use. Whether or not that's "too much" depends on:


how much your panel supplies (which also depends on how much light hits it)  
the power usage of your boost converter to run the 'duino &amp; the valve (even when neither of them are using much/any power)  
how much power thw 'duino uses (which actually depends on the software code you use)  
how much power is used by the valve, how often it's used, and how long it's "on" at one time.


All of those are things that you'll have to consjder when choosing your battery, PV panels, software &amp; components.

EDIT:

Here's a quick diagram to help with your understanding of how the battery's internal resistance works in the circuit.



Whenever there's more current coming from the charger than what the boost converter uses, there's a "backup" of current, which raises the back-EMF above the battery's current internal voltage, so current goes through the load, and through the battery's internal resistance, in parallel.  
Whenever there's not enough current from the charger, power/current comes out of the battery's internal voltage, through the battery's internal resistance, then through the load (boost converter). Thus putting internal resistance &amp; load in series, and lowers the voltage the charger sees in the circuit, since it effectively hooks in between the two "voltage divider" resistances (ignoring the reactivity of the converter, for now). Picture the "voltage divider" property like this (same circuit, just drawn differently):