ULS and SLS

The terminology is confusing you may need to check with the Geotech.

How I see it, you have 3 bearing capacities:

1. What you get out of the bearing capacity equation (this is the largest value)
2. A factored bearing capacity (usually value 1. times 0.3-0.5) - this is what you compare to structural loads i.e. Factored up structural load must be less than factored down bearing capacity
3. An allowable bearing capacity (1. divided by 3 or 4) - in modern practice you really shouldn't be using this any more.

Unfortunately the terminology isn't really agreed on and some people will call 1. 'Ultimate limit state bearing capacity' and 2. 'factored ultimate bearing capacity'. I prefer to call 1. Geotechnical Ultimate Bearing Capacity and 2. Factored Bearing Capacity.

I'm in New Zealand so YMMV. Here's how we sorta approach it (sorta, as it's still a clusterfuck of different terminologies)
I am a Structural Engineer primarily but I also have a lot of geotechnical field testing/reporting experience so that shapes my interpretations

There's demand, and there's capacity
Ultimate Limit State and Serviceability Limit State are demand terminologies
ULS > SLS in general terms as that's the obvious intention of calculating these loads
SLS is day-to-day type loads - dead load + occupancy live loads, moderate winds and earthquakes, etc
ULS is the "oh shit" load case - house crowded to the brim, mega earthquake, hurricane, whatever

When undertaking design verification we obviously need to show that our capacity exceeds our demand
In NZ we begin with 'Geotechnical Ultimate Bearing Capacity' (GUBC) which can be calculated in various ways but is general estimated from Scala penetrometer correlations
This gives us a number - say 300kPa - that is the theoretical ultimate bearing capacity of the soil
We then modify this according to the parameters of the foundation (Strip vs Pad, width, bearing depth, etc) to get a design Ultimate Bearing Capacity
This is the theoretical 1:1 capacity to demand ratio number

Now we apply reduction factors
Common practice here is to use a strength reduction factor (phi factor) of 0.5 for gravity, maybe higher for earthquake
So 300kPa * 0.5 = 150kPa - compare this directly against your ULS bearing demand
For serviceability we deviate from LRFD to working stress and use a FOS of 3 - as you put in your post, we divide 300kPa by 3 = 100kPa
We then compare this against our SLS bearing demands

The basis for using working stress for SLS is the answer to your 3rd question, sort of - the guy who did all the testing in the 70s to calculate the Scala penetrometer calculations did a bunch of work to conclude that a FOS of ~3 was sufficient to limit deflections to 25mm
So, 50 years later, we haven't shaken that demon as no one has bothered to investigate any more
Seems to work though