The Thomas Cottaine of 1700 in Lancaster County was probably the same Thomas Cotton who was the Undersherriff of Charles City County in the late 1600s, who also had land and a residence in Surry County and who had land in Prince George County. He was an immigrant from England arriving in Virginia most probably in the 1660s around the time of the English Restauration. His first wife was a woman by the name of Susanna by whom he may have had a son by the name of Robert who died in the first decade of the 1700s. His older brother, Walter, also immigrated from England in the same timeframe. Both brothers participated in Bacon's Rebellion against the govenor of Virginia, yet received pardons when the rebellion was crushed. Thomas was a contentious person and involved in a series of court cases throughout his life in Virginia. Walter married a woman named Elizabeth who was probably brought to Virginia expressly for that purpose and who was certainly much younger than her husband. Walter and Elizabeth probably had three sons (Richard, William, Thomas II) and one daughter (Anne), all probably born in the 1690s. The only child who can be definitely documented as that of Walter and Elizabeth is Thomas II. However, dna-testing has shown that the descendants of the above-mentioned William and Thomas II have to have a common male line ancestry and the only person available during that period who could serve as a common male line ancestor is Walter Cotton. Thus, even though there is no documentary evidence that William and Thomas II are brothers, it is 99% certain that they were, due to the genetically extremely rare y-chromosome structure they have passed on to their respective male line descendants.
The Richard Cotton of 1716 in Lancaster County is probably another brother of William and Thomas II, but most likely died before Thomas I in 1718. William appears to have been provided with land by Thomas I in Prince George County and Thomas II was the designated heir of Thomas I.
Thomas I took as a second wife, Mary (Bassnett?/Rogers?) Hyde, whose first husband (and father of her three surviving children) was Richard Hyde II. Mary's oldest child, Anne Hyde, was the wife of a John Smith and the mother of a girl by the name of Mary. This Mary Smith appears to have been the wife of William Cotton, the brother of Thomas Cotton II. Mary's second surviving child was a son, Richard Hyde III. Her youngest surviving child was Jane Hyde who was born in 1697 give or take one year. Jane Hyde was the wife of Thomas Cotton II.
I have long suspected that the Anne Mylone mentioned in the will of Thomas Cotton I was the wife of William Malone and was a daughter of Walter and Elizabeth. However, there is no documentary proof, only very suggestive circumstantial evidence. Interestingly, there is Anne Cotton appearing as holding 50 acres of land in Virginia from Queen Anne in 1704, if I remember correctly. If this woman is the same as the Anne Mylone of the will of Thomas I, then she would have been the oldest child of Walter and Elizabeth or a child by an earlier wife. There is also a possibility that this Anne Cotton could have been the sister of Thomas I and Walter, in which case she would have been too old to have been the wife of William Malone or the mother of a daughter born ca. 1714, as Thomas I and Walter were surely born in the early 1640s.
The Thomas and John Cotton who were the Godparents to Thomas Mullins II in 1741 would have been the two oldest sons of Thomas Cotton II and Jane Hyde. Based on what you have written, I would also assume that your ancestor Bud Mullins was a son of Thomas Mullins II. I would also assume that Bud was a nickname and that his name for formal purposes (baptism, marriage, certain land contracts) would have been Edward or some such other name.
In answer to your answer about tying people together, in Colonial Virginia, one obtained civic rights solely through the ownership of land. Thus, as long as one had land in a certain county, one was considered a resident for all purposes, whatsoever. Dual and triple residency was permitted and considered normal. Thus, though we have documentary evidence only of Thomas Cotton I having been a resident with full rights in just two counties, Charles City and Surry, but the law would have allowed him voting rights in Prince George Co. and Lancaster Co. also, assuming we are dealing with the same person. It would have also been in character as the numerous court cases he was involved in show him to have been a highly mobile person. In his later life, though, he submitted himself to the stronger will of his second wife, Mary Hyde, and appears to have lived out his old age in Surry County.
As for me, in spite of my having Japanese nationality and living in the Tokyo metropolitan area, I am a descendant of Walter and Elizabeth Cotton through Thomas Cotton II and Jane Hyde. I hope this will prove to be of help to you. If you need to contact me further, you may do so at email@example.com.