Not out of the box. you could use some washers or have some adapters made in your local machine shop if you needed but if you were looking for 1 1/4" in particular you'd be better off with the right tool

Yes

Should be ok on external bearing ones, no good for internal.

not sure about the chris king head set but i would think it should work ok. i have used both park and this one and it works just as well.

It works and it is loads cheaper. it will work with all non-integrated headsets that i'm aware of from 1" to 1.5".

Should work ok it has multiple steps on the press anvils for different sizes, if not use a high tensile washer under one of the steps. I used it to put some 1 1/2" mower bearings in recently and it worked fine.

Mate, the bushes (adapters) supplied are staggered to suit four different sizes. The first size (smallest) is Ø26.31mm. (I used this for the one inch cups). Second size is Ø29.80 mm, third is Ø32.82mm & the last is Ø37.84mm (1.489inch). The Cyclus book says for use with 1inch & oversize.

Yes it does - the various steps & even reversing the "cones" allows adaptation to most press fit installations, including not only headsets, but BBs & I've use it to press in pivot bearings as well.
A little bit clunky, but not bad at this price - Park Tool's offering has a some what eye watering price tag!