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Can anyone here state simply why evolutionists believe the earth is as old as they say?

Well, that's actually more of an issue with cosmology, but I'll bite (I'm not even an evolutionary biologist).

Becuae of the light from distant planets?

The light from distant objects attests to nothing other than the age of those objects. Light from a star that is thousands of light-years from earth only means that the star was around thousands of years ago, it says nothing of the age of this planet.

Or carbon dating? My understanding is that carbon dating has a MUCH shorter lenght of accuracy than scientists lead us to believe, (something like 4000 years, tops).

I thought that carbon dating was viable up to around 16,000 years -- but I could be wrong. In any case, carbon dating is only useful for determining the approxmiate age from remains of organic material (that is, living things) and even then only under specific circumstances (they would need to be buried in such a way as to not be contaminated later on). There are other dating methods that are reliable for a far greater time range used for non-organic minerals and these are one of the common tools used for determining the earth's age. Offhand I can only think of radiometric isotope dating, but there are others, and age is determined by a multitude of tests, not just a single test from a single rock sample.

Thanks for you answers. Reason I ask is because if one canculates the age of the Universe from how long it takes light to reach us from distant stars/planets, then wouldn't your calculations be off if it turns out the speed of light was not always 186-K? (What if it's slowing down?)

Also, geological evidence depends entirely on the assumptions we place upon it. Suppose God really DID create the earth as the Bible says. That brand-spanking-new earth would, by necessity, have SOME "geology" to it, no?

If an alien race found a manufactured golf ball, they might reasonably wonder how long it took for the white skin covering to naturally "form" on it. How many eons for that skin to become dimpled through the effects of erosion, or perhaps gravity. They might be stumped by the rubber center. Did it form first, or was it compressed into a sphere by the billions of years of gravity acting on the outer skin... etc.