Why don’t Garbage trucks spawn in cities skylines

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.

Depends. Are these assets you created yourself or that you downloaded from steam? If the latter, I would say that steam has been plagued with congestion due to the game's nodding popularity it may just take a bit longer after you subscribe. Also, when subscribing, make sure that you have the checkmark on the assets' picture AFTER you back out of its description page on steam. I had an issue where I was hitting subscribe on the description page but wasn't registering immediately when I backed out.