In Graiméar G na B C 9.58, it says:

Cuirtear an dara focal sa ghinideach i logainmneacha áirithe: cathair Chorcaí; Contae Luimnigh.

Notice how unclear this is: the second work goes in the genitive only in certain placenames like Cathair Chorcaí. Why only certain ones?

In an Bíobla Naofa, as published in Maynooth, in John 21:1 we read: Ina dhiaidh sin thaispeáin Íosa é féin arís do na deisceabail ag Muir Thibirias, agus seo mar a thaispeáin:

I suppose there is a difference between "Lake Tiberias" and "the Lake of Tiberias", but why not Muir Thibiriais?

In Genesis 31:21: Theith sé lena a raibh aige agus dʼimigh leis agus ghabh sé thar an Abhainn ag déanamh ar Shliabh Ghileád.

Mount Galaad is not slenderised for the genitive.

Does anyone have a view on this? Would all non-Irish placenames be the ones that are not slenderised in such phrases?

"Áirithe" is misleading here.
All such names are in genitive relation. And all placenames having a distinct genitive form (1st, 2nd, 3rd and 5th declension) will be declined as such.
Except they are part of noun phrases, e.g. Contae an Dúin vs. Contae Dhún na nGall



Why should Greek/Hebrew names be in 1st declension in Irish?
Koine Greek genitive of Tiberias would be Tiberiados (3rd declension, d-stem), probably.
But usage of the etymologically corresponding 5th declension form in Irish ("Thibiriad" like "carad") would be strange.

They are more likely all 4th declension in Irish, i.e. no change in genitive.

Thank you. Maybe Maynooth looked at it like that. You can also see the name Liam, which is not declined for the vs and gs in Irish, because it is a loan.

Why is it lena a raibh aige? Shouldn't it just be lena raibh aige?

_________________
I'm an intermediate speaker of the Corca Dhuibhne dialect of Irish and also have knowledge on the old spelling
Soir gaċ síar, fé ḋeireaḋ thíar

You're right, Séamus. I took that from An Bíobla Naofa online at https://www.bible.com/bible/554/GEN.31.ABN - but maybe it wasn't correctly digitised...