There are no specific rules vis a vis how a character starts combat. D&D5E attempts to avoid that kind of specificity when basic logic and reasoning will do:
- If the character is hanging out in a tavern, his hands are empty... or full of food. Utensils are Improvised Weapons, by the way.
- If the character is hanging from a chandelier, at least one hand very well ought to be weaponless (in so far as one does not wield a chandelier, it had best be making use of one hand).
- If the character just finished picking a lock, his hands are empty.
- If the character just finished picking his nose, at least one hand is empty.
- If the character is dungeon delving and carrying a torch or lantern, he's probably got a weapon in his other hand or has a shield strapped to his arm.
- If the character is dungeon delving and doesn't need a hand for a light source, he'd be smart to have both weapons out.
Either way, it's a matter of common sense and story description, not codified rules.
You will never have to worry about passwall targeting the barrier
Passwall's description states (emphasis mine):
A passage appears at a point of your choice that you can see on a wooden, plaster, or stone surface (such as a wall, a ceiling, or a floor) within range, and lasts for the Duration.
The surface of the Cube of Force creates "a barrier of Invisible force". If the caster cannot see the barrier, typically requiring a spell like See Invisibility, then they cannot target the barrier. But even if they can see it, the barrier isn't wood, plaster, or stone. Thus, it is an invalid target for Passwall as far as a direct cast.
With that in mind, the only thing we need to worry about is the cube's barrier overlapping with an opening, created by Passwall, on a different surface. In that case, Passwall's effect has nothing to do with opening a passage in anything other than the surface it was originally cast on. Thus, we don't have to worry about any effect applying to the cube, aside from the specific description in the cube's text.
The cube loses 1d6 charges because the item's description says it does, but no hole is created in the barrier because the cube didn't say that would happen, and the Passwall spell wasn't cast on the barrier.
Repeatedly colliding with a Wall of Fire will continue to drain charges
The cube's description only states:
The cube loses charges when the barrier is targeted by certain spells or comes into contact with certain spell or magic item effects
The text makes no mention of "The first time the barrier is targeted..." in contrast to many other effects that do use this language. For example Wall of Fire itself, which says:
A creature takes the same damage when it enters the wall for the first time on a turn or ends its turn there.
From this, we can determine that no matter how many times you run into the same Wall of Fire, the cube will continually lose charges, because there is no exception which states that the drain was only meant to be applied one time.
Best Answer
Consider making the tile manipulation a Lair Action.
Lair actions are a method other than Legendary Actions to give a monster ways to balance the action economy between themselves and the party of Player Characters. They usually pertain to altering or using the terrain of their lair, and can only be performed by the monster (or NPC) when they are in their lair. You've mentioned that the party will be facing this BBEG in their lair, so it would make sense here. The benefit of making the manipulation of the anti-magic tiles a lair action would be that the BBEG will still have full use of their action, bonus action, and object interaction on their turn, not needing to waste any of them.
As an example, an Ancient Red Dragon has the following Lair Action options, and chooses one to perform on initiative count 20 (losing initiative ties) each round.
Because you know you want your BBEG to manipulate the antimagic tiles every turn, you could simply limit them to one option for their lair action: to manipulate the tiles as they see fit.