Arda an Chaisleáin
caisleán, dúnfort
You don't. It is not a Gaelic name.
In Scottish Gaelic Frangag is the equivalent of Frances.In Irish it would be Proinséas.
Ballynahinch Castle
Eva is an Italian equivalent of the Irish name Éabha.Specifically, the name is a feminine proper noun. The Irish and the Italian forms translate as "Eve" in English. The pronunciation will be "EY-va" in Italian.
Woulfe's "Irish Names for Children" gives Séarlait as an Irish Gaelic form of the borrowed French name.
The name Martin can be translated as Máirtín or Mártan in Irish; the Scottish Gaelic is Màrtainn.There are not many names that translate perfectly into Irish unless they are biblical like Diego in English is James. We can translate that as it is biblical.Séamus.
The Castle is built on a large rock at the edge of the town of Cashel in Ireland. Caisleán is the Irish word for Castle and this would be the origins of the name Cashel.
In Irish it's "Buann an fhoighne ar an gcinniúint"
By "connor", I assume you mean "connard". In which case, "connard" is an insult, and can translate roughly to "idiot". If you mean "Connor" as a person's name, it would be "Conner". Names do not translate.
Names are not "translated" as such. Irish surnames have an Irish Gaelic form, but many given names used today really don't have equivalents in Irish. That said, Jacqui Ní Néill if unmarried or Jacqui Uí Néill if a married name.
Not really a traditional name in Ireland, I've seen "Seanchán" used as an equivalent. In Scotland Eòin is considered an equivalent.
The Spanish surname Castillo (also Castilo) is from the Latin castellum, meaning castle (fortified building).