Umseting - Hollendskt-Enskt - en heb je het al aangeschaft?

I'm not sure but it's an action that took place in the past and not continued, I think it should be 'and did you buy it already?'

I think it should be 'and, did you buy it yet?'

The 'already' implies that you were really quick in buying it, while the 'yet' implies that it took a little while and you wonder whether he/she has bought something by now..

I don't know about the Dutch text, since I don't know a single Dutch word, however I think the English version should read:

and have you already bought it?
or
and have you bought it yet?

Chantal's explanation is correct, 'yet' is better than 'already'.