I've actually looked into this before, and come up with some answers that aren't 100% scientifically rigorous, but they're close enough to make it plausible.
To address your issues in order:
Tunnels can be dug quite deep with only basic technology. In the quasi-medieval-Europe setting that most of this kind of fantasy takes place in, you've scaffolding, pulleys and treadmills, and wagons - everything you need to haul large quantities of rock out of a mine, using cranes, lift shafts, and rope-drawn minecarts. You can prop the tunnels with wood (which works great; when the first mechanical props came in, some miners preferred the wood because it would crack and creak to warn you when it was overloaded and close to failure).
Medieval mines did exist, and sappers would tunnel beneath enemy castles to break sieges. Depths of a few hundred metres are probably plausible with a concerted effort. If there are natural caves, or your dwarves are digging in stronger rocks like granite (which doesn't require props, because its hardness means it won't collapse if dug properly) then you could get very deep indeed. Don't forget that after a few generations of determined digging, your dwarves will have a level of mining expertise beyond anything that existed in the medieval world.
Your mention of ventilation is a good one, but not insurmountable. For this, we look at termites. A termite colony has a large, wall-like part above ground that is always constructed on a north-south alignment so that the flat side catches the sunlight. This warms the air inside, and as the warm air rises, it creates a pressure differential. The colony makes sure that the tunnels that open to the outside go from ground level to the deepest parts of the burrowsand so, stale air in the nest is pulled up and out, and fresh air is drawn in to replace it.
Your dwarves can replicate this mechanism - all they need is a source of great heat... like a set of enormous forges. Place the forges at the heart of the dwarven city, and keep digging vent shafts from the outermost tunnels (and closing off vent shafts from tunnels that are no longer outermost), and as long as the forge fires are burning, the rising column of hot air will pull fresh air through the mines. Any dwarf will tell you: a burning forge is the heart that keeps a city alive.
So the air is breathable, but it's also full of dust. Over time, as your dwarves differentiate from regular folk, they'll develop things like thicker mucous in their airways that will reduce the health impact of the dust, but in the meantime... why do you think dwarves have such thick beards? A good bushy beard and mustache is nature's dust mask. Just remember to wash your beard out periodically.
Mines used candles for light for centuries, but if your dwarves are spending a lot of time underground and have a knack for metalwork and engineering, chances are they'll come up with a few tricks a little earlier than their real-world analogues.
For a start, if they're mining coal and smelting steel, then they'll be producing coke. From that, they'll have coal tar, and if they have access to peat then they can start to produce naptha, which makes good lamp fuel. If the dwarves find the right metal salts while mining, they could start to produce mantles for these lamps, to make their light brighter and more white.
Of course, in some mines there's a risk of gas explosion when using a naked flame for light, but the key innovation of a Davy lamp is basically just a fine metal mesh, which shouldn't be a problem for a dwarf's metalworking skills. With this, you can safely use a flame lantern in a coal mine, and as a bonus, the colour and size of the flame can be used to judge the content and quality of the air in your mine. No more need for canaries!
Food is something of an issue, and it may be that your dwarves will have to maintain farms on the slopes around their mine entrance. However, there are alternatives.
There are varieties of algae and fungus that can live without sunlight; some of these were discovered in caves and underground rivers, so perhaps your dwarves found them while mining and domesticated them? These are often found around volcanic vents, using the heat as their source of energy, but your dwarves have those forges burning 24/7, so there should be somewhere close to the forges that would sustain some farms to grow algae or mushrooms. For fertiliser, well... you've got hundreds of dwarves living in a mine and it's a few centuries before anyone will invent the flushing toilet, so I'm confident you'll find something. You can supplement this with the soil you dig out from the surface whenever you put in a new vent shaft, because otherwise you risk slowly depleting some of the minerals in it, being a closed system and all.
There are fish that don't particularly need light, either because they live in dark caves anyway, or because they're used to very murky water. You could maintain a few underground lakes stocked with fish as a source of protein. The fish can eat the algae, which is good, because it's probably not very palatable to your dwarves as it is. A simple form of aquaponics might work nicely, using the algae vats to dispose of sewage, and using fish to dispose of excess algae.
Pigs probably don't enjoy living in the dark, but they don't need a great deal of space, don't mind a bit of muck, and will eat basically anything. A solid choice for underground livestock.
Fermentation works quite nicely in dark, warm environments, so your dwarves will almost certainly turn to brewing as a means to purify water. As far back as ancient Engypt and Mesopotamia, there have been beer-like drinks, some of which were quite thick, in some cases resembling a thin porridge; your dwarves could add some sort of grain or mushroom derivatives to the mix to create a strange dwarven ale that provides nourishment as well as hydration. It might be a bit of an acquired taste, but history teaches us that if it has alcohol in it, then people can learn to enjoy it.
As the years and generations go by, the folk living in the mines will naturally become shorter (to aid moving around in cramped tunnels) and stockier (to better cope with hard labour and heavy lifting). Their night vision may improve, and they'll develop an appreciation for bushy, well-maintained facial hair. They'll enjoy fish, ham, and their own variety of strong, strange-tasting beer. Their cities will be surrounded by mines, and they'll place great importance on keeping the forges burning.
Sounds just like a proper fantasy dwarf to me.