So, to summarise what we know so far and some more information on two of the tunes.

Tune # 1 Codiad yr Ehedydd: The Rising of the Lark

Composer:
OWEN, DAVID (‘Dafydd y Garreg Wen '; 1711 - 1741), harpist attained fame both as harpist and as the reputed composer of the airs called ‘Dafydd y Garreg Wen’ (‘David of the White Rock’), ‘Codiad yr Ehedydd’ (‘The rising of the lark’), and ‘Difyrrwch gwyr Criccieth’ (‘The delight of the men of Criccieth’). He died on 2 August 1741 and was buried in Ynyscynhaearn churchyard.

"Codiad yr Ehedydd: The Rising of the Lark published by
Olivia Buckley Dussek , 1799-1847. Single work for solo harp. Three of Olivia Buckley Dussek’s arrangements of Welsh Airs - The March of the Men of Harlech, Rising of the Lark and Megan’s Daughter have been re-set and re-published by Adlais as part of our ‘Victorian Favourites' series."
   
The Traditional Tune Archive has some information including the music notation and ABC Notation.

"Codiad yr Ehedydd"

Tune # 2 Begone Dull Care

Begone Dull Care notes the following printed sources:

Printed sources : - Aird (Selection of Scotch, English, Irish and Foreign Airs, vol. 4), 1796; No. 173, p. 65. Howe (Complete Preceptor for the Accordeon), 1843; p. 8 (appears as "Kinlock"). Kerr (Merry Melodies), vol. 3; No. 399, p. 44. O'Farrell (Pocket Companion, vol. II), c. 1806; p. 159.

Begone Dull Care has this comment about Benjamin Britten's arrangement for performance in a school environment:

"The text was written by an anonymous artist in the 17th century, and was published in a book called English Lyrical Verse (The King’s Treasuries)."

Tune # 7 See John the Clockmaker's post on 19 May 20 - 06:46 AM
"Tune is "Illsley" and was written by John Bishop (1665-1737)."

I'm wondering whether one of the printed sources mentioned on the pages I found could also hold the other tunes that we are looking for. Further research is needed. I'm hoping to find online copies to see a list of other tunes in the books or even digital copies of the books to be able to see the music notation of other tunes.