Translation:We have lived together for sixty years.
why should "al" not be translated with "already"? as in "we lived together for already sixty years"
In English, "already" needs to be in a different position in the sentence. So you could say, "We have already lived together for sixty years".
Is it correct for this sentence to use the present perfect? Yes, in English, one would use the present perfect here; but since we're referring to an ongoing state, I've been under the impression that Dutch would instead use the present:
You are quite right. The same thing is mentioned here: http://floatingindreams.com/2012/06/09/dutch-vs-english-past-simple-vs-present-perfect/. If this implies that they are not living together any more, past perfect should be used and past perfect continuous at that. I am confused now too.