are these warm enough for winter or are they "for show" boots?

These are not winter boots (though some Palladium boots are), so they are not particularly warm. Also I wouldn't call them show boots. I've worn mine almost every day for close to a year now, including lots of walking around cities and some hiking. They look practically new, quite durable.

How tall are the boots from the bottom of the heel to the top? Which color is closest to Coyote Brown?

About 5 1/4 in. from heel to top. The dark khaki, I would say, would be the closes to coyote brown.

how well do the rubber soles handle extreme heat? do they melt as converses do, or are they of a sturdier rubber?

Hello! I've worn these boots in tropical climates, Samoa and Hawaii, especially, and in the high heat of Texas with no trouble. No melting.

Are these boots waterproof?

No, but you can buy a spray on water-proofer if you need them to resist water -- they still won't be 100% waterproof, but they will stand up to rain and puddles that aren't deep enough to submerge the tongue and/or lace eyelets.