We, unfortunately, never have enough information about the Dwarves in Tolkien's books. We mostly see them from the outside points of view - either elven in Silmarillion drafts, or hobbit one in the Hobbit and LOTR.

There is some information in those books, however, that lets answer some of your questions.

First off, I do not think dwarves would just hollow out a solid mountain. Wherever we have mentions of underground structures, they are based on some natural caverns. Nargothrond was built by the elves in the natural caverns that were fist inhabited by the dwarves. Menegroth, it seems, was also build by dwarves for the elves on the basis of the natural cave. As far as I remember, the river Celduin flowed from inside the Lonely mountain - I always assumed it looks something like Lod Cave in Thailand before the dwarves repurposed it into a city.

There is also a telling passage in the LOTR, where Gimli speaks about the caves of Aglarond and how the dwarves would work on transforming these natural caves to a dwarven city if he had his way.

Overall, the idea of a fortified underground 'burh' is a pretty popular one in the Middle-Earth, for both elves and dwarves. The halls of the Elvenking in Mirkwood are in fact a lesser version of Menegroth - a fortified underground dwelling that serves as a kingly castle, with most of the population actually living in the forest outside. (Another staple of an underground sinda-dwarven 'burh' is an underground river flowing from it).

Speaking about the food supply, we actually never encounter the idea of a completely enclosed self-sufficient underground dwelling, neither for elves nor for dwarves. The only completely self-sufficient city without any links to the outside was Gondolin, and it was not in a cave system, but in the open space surrounded by the mountains. All other underground burhs - both elven and dwarven - had their gates open most of the time. For example, I remember Moria closing it's western gate in the Second age, but it seems their eastern entrance stayed open all the time.

The dietary requirements of the dwarves seem to be the same as the other people of Middle-Earth - they eat their food for extended periods of time, seem to like it and never display a need to eat some particular product to refill a microelement deficiency. (Actually, elves, humans, hobbits and orcs are biologically the same species according to Tolkien himself. Dwarves are a close copy of the same design by the other maker, so it seems they are still a carbon-based lifeform, although not genetically compatible with everyone else.)

So it seems to me it's not a stretch to assume that dwarves actually farmed and raised animals (say sheep and ponies) outside of their cave cities. Unless they could find a settlement of other people nearby and establish trade relationship - Erebor and Dale, for example, had it all worked out perfectly before the dragon appeared.

As for the light and air - we all remember the description of the air shafts in Moria. Presumably, dwarves were good enough engineers to calculate the neccessary number of shafts and their ideal locations to ensure the proper air supply. Having big enough openings for underground river flow in and/or out, as Tolkien's underground cities often have, would also help with air circulation. We also have mentions of some sort of artifical lighting in the song of Durin, as aready discussed before.

As for agoraphobia and claustrophobia. I think the movie indeed does overdo the 'gaps' a lot. Mostly the dwarven cities would be differntly sized caves, connected by artificial corridors. All the dangerous gaps would be spanned by bridges, separated by railings, etc. Now, claustrophobic dwarves would have it bad, though.