I shall put this very bluntly at the outset:
There is no evidence of the parents of Edmond Lewes of Lynn, Massachusetts, who came to Massachusetts Bay Colony in 1634 from Ipswich, Suffolk, England. There is no evidence that Edmond Lewes of Lynn, Massachusetts is the same as Edmund Lewis of Llys Talybont, Glamorgan, Wales. There is no evidence that Edmond Lewes had any Welsh connection whatsoever.
There is no evidence of the surname of Edmond's wife, Mary. The claim that Mary's surname was Carey of Carew is from an unsupported entry in the LDS genealogical database. Therefore there is no King's Lynn or Lynn Regis connection in Norfolk.
If anyone has evidence to support these claims, I would be delighted to see it!
The Arundel Castle connection is from 1602 - WILLIAM LEWIS - 1671 of Stoke-by-Nayland, England and Some of his Ancestors and Descendants by Isaac Newton Lewis, published in 1932. The connection of William Lewis of Roxbury to Edmond Lewes of Lynn is not established in this volume. He merely states, without evidence "Edmund Lewis, cousin of William." I have searched the parish records of Stoke-by-Nayland and I did not find record of Edmond Lewes's birth, marriage nor the birth of Tomas and John.
Lewis is not a Welsh surname. The Welsh Lewis surname comes from the Anglicization of Llewellyn when Henry VIII demanded that all Welshmen adopt surnames in the 16th Century as by the English custom. Thus a Welshman known as John ap Llewellyn (John, son of Llewellyn) became known as John Lewis. (See Late Sixteenth Century Welsh Names by Talan Gwynek, http://www.s-gabriel.org/names/talan/talanWelsh16.html)
Lewis is, in fact, an English surname, derived from Saxon and Frankish roots, coming to England as Ludovich or Ludwig. It's transliteration occurred in England, most probably in East Anglia, during the Danish and subsequent Norman occupations. Lewes in England is much older than Lewis in Wales.
This does not imply that there are no Welsh Lewis lines in America, just that not all Lewis's are of Welsh ancestry.