Ath Cliath section
We have a bit of a problem here as the Irish name Baile atha Cliath was first written down in 1368, in the Annals of Ulster. The other sources mentioned such as the Annals of the Four Masters were written down even later.

In other words, we don't know what it was called before it was called Duiblinn / Dyflin, and to name a whole section "Ath Cliath" is fanciful at best. I'm renaming it "Pre-Viking Dublin" unless anyone can come up with better. Some will say it is "likely" that it was called BAC before 795, others that it is just as likely that the Vikings copied Dubh Linn.