Is there a way to implement simple formula execution like in a spreadsheet?
Alternatively, is there a way to have one of the Private Window buttons display something like the Spreadsheet piece property that can be private and unique for a given player?
Is there a way to count up all of the pieces of a given type on the board?
Thanks, I know I’m a total noob. 
You can increase / decrease the value of a property but you can’t do much as far as formula’s beyond that as Vassal does not have a math engine at this time. Newer versions of Vassal should have a math engine in it.
Yes. This would probably involve the use of Global Properties and/or Dynamic Properties to track values. You can then set up something in a private window to display those values.
Yes. You can have all the pieces you want counted to have the same Marker Trait. Something like UnitType = Army. You can then send a command to all pieces with UnitType = Army and have them all increase a Global Property being used for keeping track of the unit count. (something like ArmyCount)