I think this would be a great item for a Sun Soul Monk.  Radiant Sun Bolt can't pass through face number 4 because it is categorized as a spell attack, but Searing Sunburst still can.

What classified as "Spell effects"?

It says that it “starts with 36 charges” instead of it “has 36 charges”. Does this mean it can have more than 36 charges?

Any effect that is caused by a spell

Assume that 36 charges is also it's cap. 

If I activate the Cube of Force while I'm inside a creature's esophagus, does the cube fully expand to 15ft and cause damage to the enemy?

no, it would just cause a barrier to appear surrounding the creature, who would be unharmed to my understanding

What if the creature is larger/longer than 15ft and the barrier bisects it in some way? 

It says certain objects can't pass through, could they pass through if they originate from inside the cube? Like if I shot a creature outside of it with an arrow while say, side 5 was activated, would the arrow hit them?

Quote from Oculusthehalfdragon >>

It says certain objects can't pass through, could they pass through if they originate from inside the cube? Like if I shot a creature outside of it with an arrow while say, side 5 was activated, would the arrow hit them?

 No. It's impenetrable by the specified things, regardless of direction.

Quote from Gibruis >>

What if the creature is larger/longer than 15ft and the barrier bisects it in some way? 

No need for it to be larger than 15 feet:

"If your movement causes the barrier to come into contact with a solid object that can't pass through the cube, you can't move any closer to that object as long as the barrier remains." (emphasis mine) 

Only your movement is proscribed; the cube could bisect something that cannot pass through it, but you can't move closer to that thing, & it can't penetrate the cube (any farther than it already is).

So on one hand, I'm tempted to say that if you were in a grassland, surrounded by grass spaced less than 15 feet apart: You could use this item's 3rd setting to block any living thing from coming or going, but you wouldn't be able to move (Except to jump\hover straight upward. You'd "land" 5 feet above the grass tips, upon falling; If you can't hover or fly, you're now even more stuck, >5 feet off the ground with no traction.)

If you were within a 21 foot cavity inside the body of a gargantuan plant or creature, standing next to one fleshy wall, atop the nonliving deck of a ship, & activated its barrier to block "living matter", you'd be able to move away from the flesh but not towards it. 

The muddy definition of "object" may also come into play here. (Crawford has stated that Dragon's breath is neither an object, nor a magical effect, & the Dungeon Master's Guide's definition specifying 'Objects' as "discrete" objects, is followed by listing things like a whole cart, & a wall. ... So... 🤕)

If you are standing inside a creature, right on its stomach lining, I believe you could walk around inside it (as long as you don't jump) because you're not moving "any closer" to the creature you're already touching. (If at any point you use your movement to move any distance (even 12 inches) off it's surface, you're dangling with no traction & can't touch it anymore.) 

HOWEVER, it does say "living matter can't pass through the barrier", so unless you're strong enough to rip free any tissue bisected by the cube, or drag the whole creature along with you, it could be interpreted that you are unable to move too. I could be persuaded to this position because it fits the wording... but it seems less fun. 

Either way, you could effectively grapple medium or smaller creatures with no saving throws, by moving 7.5 feet from their position & hitting 3 (or 5). 

From the creature's perspective, it has an impenetrable hard edged box inside it, pinning any tissue it bisects. (Try activating 7.5 feet from its heart?) 

Perversely, under either interpretation, a 20 foot cloud of flies would pin you, on setting 3. (Plus there are flies in there with you.) 

Contact with fog, mist, rain, water, & even lava would not prevent your movement, on any setting, because they are not considered "a solid object". Any settings besides 2 & 5 would allow you to drown, & even for those you need to start with your head above water; in which case the cube will be buoyant! 

Incidentally, with setting 5 you'd be in total darkness, unless you've got a magical light inside. Light, heat, sound, nothing will pass through. Definitely no moving except in midair, because even if "the ground" is not an impediment, every loose pebble, shrub, & blowing leaf is almost unquestionably a "discrete object", & will stop your movement in its direction.

So a few questions if I press the 6 side:

1. 
If I cast a spell at a point outside the sphere, will it work? Say Wall of Fire, Banishment, or Synaptic Static. The spell doesn't go through the cube, it starts outside it. Thoughts?

2. 
It says "If your movement causes the barrier to come into contact with a solid object that can't pass through the cube, you can't move any closer to that object as long as the barrier remains." What if there's a creature already inside the cube? Like if I'm there with an ally. Do they move with me?

if you click something like the 4th face, doe the effects from the previous faces also activate? Like does it take the other effects when you press a higher face.

You can change the barrier's effect by pressing a different face of the cube and expending the requisite number of charges, resetting the duration.

I'd go with no, pressing side 4 means you only get the effects of side 4.

anything that's considered a spell

1. RAW I think you can target spells inside as well, it's not a spell effect... and vice versa from inside

As a wizard, can I cast a concentration spell, then on my next turn hit side 5 and effectively remove myself from the combat while still concentrating on the spell?

yes

So, if you choose " Nonliving matter can't pass through the barrier.", you wouldn't take falling damage if jumping/falling off a cliff. Neat!

The cube specifies non-living matter; does this mean my necromancers zombies could pass through that barrier?