What’s the smallest tank I’d need to inflate a fat bike tyre?
I’d like one of those vertical ones as I don’t have a lot of room to store it.
Do they generally come with a hose and connectors?

The bigger the better as more capacity = more time for the tyre to seat as you push, squidge, bounce, etc. A litre and you will need A1 conditions or luck but twenty five litres will be more than enough. What size you lookin at?

I’m running tubes so seating is not such an issue.
The main reason for buying it is to blow out the dust in the cooling elements on our chocolate cabinet.
That won’t need too much puff but if I can also pump up a tyre that will be handy. So ,smaller the better.

Can’t you just suck the crap out of the cooling elements with a Henry/Dyson/Miele whatever? If you use a compressor it’s going to blow the [stuff] all over the shop!

I hoover it every week. The main came to service it and indeed the world’s biggest cloud of dust came out of it.
The ladies weren’t happy!
So a regular blast will keep it clean and running cool.

Do they generally come with a hose and connectors?

Yes, and loads of crap accessories, but not a tyre connector. I cut one of an old track pump and joined it to the hose using a P-clip thing.

I’m running tubes so seating is not such an issue

Well you can manage without a reservoir/tank for that as you don’t need the sudden rush of air. Buy whatever you want for blowing out your dust and consider it a bonus if it will pump up your tyres. The tank is there to stockpile air for a bigger whoosh than the motor can provide

but not a tyre connector

Most come with a car tyre (schrader valve) inflator just nothing to fit a presta valve