Way to unlock features right off the bat

Districts don't actually do anything to restrict or direct services. Services will be user for whatever is in their local area, regardless of district boundaries (exceptions for power and water, as you distribute them manually).

Districts are useful for:

• industrial specialisation (agriculture, forestry, etc)
• policy specialisation (education boosts, tax cuts, etc)
• controlling lorries (technically a policy specialisation, by way of the heavy traffic ban, but this is important enough to warrant its own entry!)
• adding personal flavour to your city.

On that basis, the game doesn't care what districts your fire stations and fires are located in, which is why this can happen.

As to why it sends fire engines from a distant station instead of a local one (and thus, how you can prevent it), well... I'm not sure, but here's a few thoughts that may help:

• Are all the engines from the closer stations in use?
• Is that road in the middle, where the fire is, one-way in the up direction? If so this makes it "further away" from the stations on the right that it seems, and would be why it's a greater fire hazard than you might expect. It might also be confusing the distance calculation (and thus the selection of station) - consider moving one of the stations on the right downwards.

I think you've got it right, it really is that simple. A single bus depot is sufficient to supply enough buses to a city of any size¹, and its position isn't really important as long as it is connected by road to any bus routes you create (if your city isn't entirely road-connected you'll need a depot in each area you want bus routes in). 

The only other thing to be concerned about is that depots create 50 noise pollution, so you probably don't want them nestled into the middle of your residential areas.

If you care about fine details, buses will need to travel from/to the depot to get to/from bus routes, this only happens when routes are created/edited/deleted (as you say, they never rest). This means new bus routes set further away from a depot take longer to start working. Also, if you change your city's bus funding (changing the number of buses on all of your lines), you might find a swarm of buses around your depot. These are temporary issues that you can probably ignore.

_{Also, I guess if you only have one depot and it catches fire you might have a problem, but meh.}