"Ce la siamo cavata" means "We succeeded".
You could also say: "Siamo riusciti a cavarcela".
"Cavarsela" is a particular kind of verb. In italian we call it "verbo pronominale" o "verbo idiomatico".
It is formed by a verb followed by 2 pronoun particles.
The meaning of these verbs is different from the meaning of the "simple" verbs:
- "cavare" has a different meaning from "cavarsela"
- "prendere" has a different meaning from "prendersela"
and so on.
When you coniugate the verb, "la" remains the same, and indicates the direct object of the verb (your second question: yes).
On the contrary, "se" follows the person of the verb.
For example:
io ME la sono cavata - sono riuscito a cavarMEla
tu TE la sei cavata - sei riuscito a cavarTEla
lui SE la è cavata - è riuscito a cavarSEla
noi CE la siamo cavata - siamo riusciti a cavarCEla
voi VE la siete cavata - siete riusciti a cavarVEla
loro SE la sono cavata - sono riusciti a cavarSEla
Finally, to answer at your first question, you find "CE" instead of "CI" because, when it comes before "LA", "LO", "LI", "LE", "NE", "CI" becomes "CE".
So yes, "CE" is the reflexive, and it's not "CI" because it is followed by "LA", and in this case "CI" becomes "CE".