Is it possible that Earth had a shorter orbital period at 3000 BC

It's interesting to me that the ancients, such as the Egyptians, Babylonians, and Greeks, divided the circle into 360 degrees. As far as I am aware, the oldest civilization we know of appear to be the Sumerians, who maintained the Sexagesimal numeral system and were keen observers of the skies. Is it possible they based their numerology on the 360 day year they observed during their time? If not, how far back in time would we have to go when the orbital period of Earth took 360 day

Virtually all ancient civilisations which have left records were thoroughly aware that the year had about 365 days, after earlier estimates of 360. The difference is that primitive eyeball observation over one year is not good enough to show the difference and is therefore error-prone. When techniques improved a little, or they measured over more than one year, the extra five days became obvious. If the measurement was taken over say three years, the difference would be 15 days and obvious. By a hundred years BC, the estimate had improved to 365.25. 

Since culture nearly always lags behind science it took a while for the mass of people to recognise this fact. 
Pope Gregory 13 introduced our present calendar in 1582, but the British did not officially adopt it until 1752 and the last European country to adopt it did so in 1923. In some places it was rejected because it had been proposed by a Roman Catholic pope. (Compare this with the people who reject evolution on religious grounds.) 

In reality, days have been getting longer over periods of a million years or so because the Moon is slowing the spin of the Earth. A thousand or so years ago, a day would have been about 2 seconds shorter. This change is not uniform but estimates are that the day was about 21 hours long when animals first appeared on Earth, so then there were more days to the year, not fewer.

The Earth is actually moving away from the Sun. But it's doing so at a rate of about 1 micrometer a year. That would mean that the orbit of Earth has increased about 5 millimeters since 3000 BC. Not enough to have added 5 days to the orbital period. 

You are on the right track, though, for why a circle has 360 degrees - It has to do with the travel of the Sun through the sky. The calendar then was 360 days long, and they didn't have precise instruments like we have to day to measure the exact travel path of the sun. They just knew that in the one calendar year (360 days), the sun made a complete journey through the sky.

No... here's why a circle has 360 degrees.... 

In 1936, a tablet was excavated some 200 miles from Babylon. Here one should make the interjection that the Sumerians were first to make one of man's greatest inventions, namely, writing; through written communication, 
knowledge could be passed from one person to others, and from one generation to the next and future ones. They impressed their cuneiform (wedge-shaped) script on soft clay tablets with a stylus, and the tablets were then hardened in the sun. The mentioned tablet, whose translation was partially published only in 1950, is devoted to various geometrical figures, and states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60 + 36/(60^2) (the Babylonians used the sexagesimal system, i.e., their base was 60 rather than 10). 

The Babylonians knew, of course, that the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle, in fact that was evidently the reason why they chose to divide the circle into 360 degrees (and we are still burdened with that figure to this day). The tablet, therefore, gives ... Pi = 25/8 = 3.125 - which we know as a more accurate ratio of about 3.141592....