Some other information that has come to light is:
Catherin Jones' (wife of William Davies) parents were -
Richard Jones 1855 b Aberdaron, Caernarvonshire
Jane ? 1853 b LLaniestyn, Caernarvonshire
I think Jane's maiden name was Davies but I still have to confirm that. Richard and Jane settled in the Llanllyfni area and their children were -
Mgt (Margaret) Jones Dau 1875 Caernarvonshire
Hugh Jones Son 1878 Llanllyfni. Caernarvonshire
Humphr(ey) Jones Son 1881 Llanllyfni. Caernarvonshire
Anne Jones Dau 1883 Llanllyfni, Caernarvonshire
Catherine Jones Dau 1885 Llanllyfni, Caernarvonshire
Elizabeth Jones Dau 1887 Llanllyfni, Caernarvonshire
Griffith Jones Son 1889 Llanllyfni, Caernarvonshire
Thomas Jones Son 1892 Llanllyfni, Caernarvonshire
Mary Jones Dau 1894 Llanllyfni, Caernarvonshire
It would seem that the family moved to Llandwrog from Llandudno and were there at least from 1916 to 1920. Apparently William was still alive when the family moved to Penygroes. Catherine passed away in her daughter Jane's house. William and Catherine are buried together at Machpelah cemetery Penygroes and all that marks the grave is a cross with W.D on it.
Gina also remembered that Teddy (Edward) Davies was a boxer, and apparently quite a good one.
I am hopeful that Gina will be able to talk to the extended family and pass on some additional information.
One intriguing thought I had was that of the boot/shoemaker shops listed in Llandudno, one was owned by a William Jones. The Rates Book image you kindly posted shows that the other family living at Vron Cottage was a William Jones. I wondered whether there was a connection between William Jones and Catherine Jones? Was this how they met? Another thread to explore.