Rhys Ap Thomas/Katherine Howard, NOT!
Your question on the proof and disproof of the Rhys Ap Thomas ancestry for Deacon Edmund Rice is very appropriate.The disproof is by none other than Donald Lines Jacobus and was publishe in The American Genealogist (he was former editor and owner of TAG) before 1933.I don't have the original TAG article but a reprint of it by the Deacon Edmund Rice Association.They are one of the best known family association with a high degree of interest in well researched genealogy.
Everyone confused on this subect should read the Jacobus reference.The disproof is four typed pages and is quite scholarly.It refers to the "Howard ancestry" as being found in "By the Name of Rice" by Dr. Charles Elmer Rice, Alliance, OH 1910.Evidence is not cited by Dr. Rice and certainly not baptismal or marriage parish records.Dr. Rice claims that Deacon Edmund Rice was born in co. Buckingham without citing his evidence.This was probably a guess based upon finding baptisms of Deacon Edmund Rice' children in co. Buckingham.I have a note that the baptism record for Deacon Edmund Rice has been found in Stanstead, co. Suffolk.
The latest newsletter of the Association further comments that Gary Boyd Roberts, FASG, also completely discredits the often reported ancestry and strongly supports Jacobus' research.
The Association still is searching for more information on the ancestry but you can safely ignore the Rhys Ap Griffith/Howard lines.
The ancestry of Thomasine Frost is well researched and you can give it some credence but I don't think there is a royal line there.