A very quick answer.
You can have a single entrance big enough for entry but rock is um, prone to cracking. So you would be able to have several to hundreds of airvents.
Completely natural caves often have small openings, or a series of cracks etc, allowing air and water and earth and small animals inside.
This would definitely help with the bad air issue that @Murphy raised.
It would also give some story to your lost explorers as they follow what they think is an airdraft leading them out only to bump into a very solid looking wall of stone.
EDITED FURTHER ANSWER
As you are a film-maker I'll just quickly go into the basics of air movement incase it has been awhile since high school geography.
Just so you know, this is just an assumption how things would work underground. I'm also writing on the fly so if anyone spots anything mixed up, please correct me!
Air moves from areas of high pressure to areas of low pressure. That pretty much means, cool and cold air sinks, and warm and hot air rises. This is due to cold air being denser and 'heavier' than warm air.
Once you have a flow of air moving from one location to another it can form a sort of vacuum effect pulling air along with it. This helps reinforce the movement. This is what you hear on the weather reports as a pressure gradient.
This movement of air can happen on small scales. Hot smoky air of a room on fire means you hug the floor for the cool fresher air. This could be why older styles buildings had double height ceilings as a natural form of air conditioning and ventilation. I do know the Minions (spelling) of ancient Greek era used this in their building design.
In your caves
You have your cave entrance. This area is cooling but still has some heat from outside. Further inside, with no sunlight reaching it, it gets colder and colder (so air will sink). Any underwater lakes or groundwater will increase this temperature difference, especially if it is glacial water. This is what you experience when you go down into sterkfontein and cango caves. Near the entrance you have the welcome relief from the hot sunshine outside but then as you get further and further in, you begin regretting leaving your jersey in the car.
You have to take into consideration the temperatures outside due to the day/night system. Hot days, cool cave entrance. Air won't move far, as at the very near entrance the air is rising, and further back, the air is sinking creating a 'buffer' plug type area. During the night, you have potentially cold outside and cool inside, this could allow a case of a strong pressure gradient and allow air to be 'sucked' inside.
Further down (speaking 1km and more) temperatures underground actually increase. This is because you are getting closer to the earth's hot centre. I think it's on the order of 25 degrees Celsius increase for every 1 km you go down. The mechanism is called the geothermal gradient, or something like that. This is why the deep gold mines are so hot. You never see a picture of goldminers working with their shirts on.
Mining note: the deep mines of anglogold have artificial ventilation to reach the depth they work at. But I don't know if this would be necessary for a natural system where man wasn't pumping the air full of dust and noxious gases. The ventilation is also to make the working temperature conditions a little bit more bearable.
So now you have hot air rising from the depths (I would imagine that it would not be very fresh or particularly healthy air to breathe) and you have cool fresh air sinking from the surface. How deep can the fresh air go?
This will relate to the cave system itself. If it's a particularly level cave system, the transfer will be slow and probably prone to stalling. If it is fairly steep, air is more likely to reach further by sinking quicker while the strong evening pressure gradient exists.
Some cave systems are very long but don't go very deep. While others, go very deep over several levels (like yours).
If the cave is very twisty and goes up and down levels, this will create a U-bend situation. Underwater lakes can also block the passage of air. Combined with natural gases etc this creates those bad air pockets mentioned before. This will limit the influence of fresh air from the surface.
That doesn't mean that there isn't any circulation below these U-bends. It just means that it will be a closed system, moving only the air already underground.
So I think, to answer your question. You have to decide how twisty your cave is, how many blockages exist to prevent air moving, and then any temp dynamics that can move the existing air around those cave systems.