My Troddens came to Philadelphia from Leeds, Yorkshire, England in 1905. Before that, they were in Navan, Co. Meath, Ireland. The name is supposedly rare in Ireland, although there were several Trodden families in the parish of Lisnadell, Co. Armagh in the Griffith's Valuation of 1859. Today, there's a concentration of Troddens in Lanacashire, probably as a result of a flight from Ireland during the great famine in 1845-1850.
McLysaght in his SURNAMES OF IRELAND claims that Trodden is derived from O'Treodain. There are Troddens in Ayrshire, as you've noted, and they appear to be of the same family as those in Ireland, but I have no substantiating evidence.