While doing some studying I came across the following sentence:

"Nuair a chuireadh sé tús le rud éigin, dhéanadh sé a dhícheall é a dhéanamh go maith."

I can see the endings for the passive/autonomous (past tense) on CUIR and DÉAN, but what I don't understand is the subject pronoun SÉ after each verb!

I thought the whole point of the passive/autonomous was that there is no subject!
what is going on here and how would I translate this sentence,

Go raibh maith agaibh, a dhaoine cumasacha!

Here it isn't simple past autonomous but imperfect (past habitual) third person.

chuireadh sé = he/it used to put
dhéanadh sé = he/it used to do/make

Please note the lenition in chuireadh (past autonomous is always without lenition: a cuireadh) and dhéanadh (past autonomous would be rinneadh)

Nuair a chuireadh sé tús le rud éigin = When he used to start something
dhéanadh sé a dhicheall = he used to do his best
é a dhéanamh go maith = to do it well

Remember:
Chuirinn = I used to put
Chuirteá = You used to put
Chuireadh sé = He used to put
Chuirimis = We used to put
Chuireadh sibh = You (pl.) used to put
Chuiridís = They used to put
Chuirtí = One used to put (autonomous form of past habitual)