Is Business class as good as first, or is first still king? also does any airlinehave 3rd class?

Since they charge more for first it must be better. Well I suppose it depends what you're flying on. On a long haul flight with designated first, business, and economy sections, first is going to be above and beyond what you get in business -- sometimes with a great degree "more", sometimes with just slightly nicer and more spacious amenities. That doesn't mean business is bad, it's just not first.

However, often times on shorter flights or smaller planes there is less of distinction, assuming there even if a business class on the flight. Either way I'd rather be in First but if Business is what I book, I'm not likely to upgrade it myself.

On planes with three classes, first will be better than business - perhaps a little, perhaps a lot. That's the only thing one can state with certainty. (Class names are not standardized. What you refer to as "third" is generally called "coach" or "economy." Some airlines, like BA, have "cutesy" names for the classes.)

Across airlines, there are no standards. One airline's business class may be better than another's first. One airline's first may be better or worse in different markets: AA's domestic F (in all but a few 3-class transcon flights) isn't up to its overseas business class, let alone overseas F. The "business class" seating of most European airlines within Europe - typically economy seats with the armrests shifted toward the middle, to increase seat width a few inches at the window and aisle, and what's left of the middle seat not sold - are nowhere near what they sell as trans-Atlantic business class. To make things more complicated, the same seat may be designated "first class" in some markets, "business class" in others.

Finally, airlines like UA may divide one nominal class into sub-classes with different seating (e.g., more row pitch) or other amenities.

In a competitive market with no universal definitions and everyone looking for a more effective way to do sell, you have to look at each case individually. There are generalizations, but no rules.

The closest thing to a rule is that F has no middle seats, while business may have middle seats (see United's 2-2-2 domestic F vs. 2-3-2 business on their 777s).

Of course someone will find an exception to this "rule".

Another "rule" is that if you buy an international business ticket with a connection to a different domestic US carrier, you will be sitting in coach, as you don't have a first class ticket.

True. Except in the "newer" business classes, the middle seats are becoming a thing of the past.

That's certainly not the case, in fact I've always flown into the US on a business ticket and always flown the domestic segment in what is comically referred to as first class.

I think that what ralfp meant was that--while business class transfers to 1st class in US domestic--if you change airlines, you tend to get the lesser of the two. If you're flying the same airline or a partner of the original airline, it's different.