Earth is NOT 6000 years old, but much older. There are MANY ways of dating Earth, and most of them are consistent with each other -Earth is about 4-5 billion years old. Uranium dating, potassium dating, rate of tectonic plates motion, rate of cooling of Earth, crust thickness, fossile ancient life and ancient geological events records, rate of mountains deterioration, records of air composition in antarctic ice, records of magnetic field reversal, etc etc.
All those records and dating technics give reasonably consistent with each other numbers about Earh being 4-6 billion years old. (Younger earth is inconsistent with many ancient fossile minerals still existing today and most of other geological records, while older Earth is inconsisten with the age of Solar system itself).
About carbon dating. In 11 century there was a monk who made Jesus shroud and everyone knew that then. With time it was forgotten, and recently the shroud was carbon dated by 3 independent labs. They cut pieces of shroud, burned it and analysed C14 content in CO2. Remarcably, all 3 labs gave same age - about 10 centuries plus/minus only few years, and that is when it was likely made by the monk. (Of cource, there are plenty of people who believe that it was made 2000 years ago and Jesus was indeed buried in it, but this is another story.
So, scientific methods of dating are accurate enough to completely rule out 6000 years ago creation, and even 6 000 000 or 600 000 000 years ago one - such recent creation simply contradicts to all other evidences badly.
About length of a day and of a year. Year length is strictly defined by Keppler's law of cubes/inverse squares, T (years) = R(astronomical units)^3/2. Say, Jupiter is 5.2 times futher from Sun than Earth, thus it's year is 5.2^1.5 = 11.8 years. So, any essential deviation in Earth orbit radius (needed to reconsile Bible with geology) would result in such changes of climate due to less (or more) heat from Sun, that it would leave tremendous geological evidence which we do not have. Nor do we have evidence that period of rotation of Earth was tremendousely different on the past than now. We do have indications that is was slightly shorter in the very distant past (a few hours shorter than now) and that Earth rotation is gradually slowing down (likely due to tidal friction with the Moon which results in Earth pushing Moon to higher orbit at the expense of slowing down itself to conserve total angular momentum). But there are no evidences that day was ever 10, 100 or 1000 shorter or longer - again it would leave so deep geological evidences due to climatic and ecological changes that would be impossible to miss. Not to say, would require extraordinary interaction with some celestial body to slow down/accelerate Earth with such extraordinary force.
So, there is no way to stretch Biblical days to such a degree that they would resemble a reality of Earth's geological and ecological past.