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Имя при рождении = Name at birth.

Yep, literally "при этом" = "along with that".

Кот воровал, при этом он прятался. The main action is "Кот воровал", the accompanying action is "он прятался".

There are many meanings listed below (from Lingvo). But I'd say that the general point is the following:

A при B (B is in prepositional) = B is the main big object and A is a small object...