Learn Portuguese (Portugal): Sou flexível e adapto o projeto ao cliente. - I am flexible and adapt the project to the client.

You could say adapto o projeto consoante o cliente (I adapt the project depending on the client). When used this way, consoante acts as a preposition and is placed directly before the noun phrase, replacing the preposition a that is normally required by the verb adaptar.

Why use sou instead of estou to say "I am flexible"?

Using sou (from ser) implies that flexibility is a permanent trait of your personality or your core business model. If you used estou (from estar), it would mean you are only temporarily flexible right now, perhaps just having a free spot in your schedule today.

I sometimes see the word for "project" spelled as projecto. Is projeto the correct spelling in Portugal?

Yes, projeto is correct. Under the modern spelling agreement (Novo Acordo Ortográfico), silent consonants were dropped from written European Portuguese. Because the "c" in projecto was not pronounced in Portugal, it was removed from the official spelling.

How would the end of the sentence change if I were adapting the project specifically to a female client?