Door doesnt mean door but thru or by. In this case it comes with the verb 'doorzetten' which means 'to persevere' or 'to persist' When you conjugate (is that the correct word) the verb you can split 'door' and 'zetten' into seperate entities.
What it actually says (or rather what anarres probably ran thru the interprator):
I felt jealous about not being able to read the thread, but I persisted and ran this thru a translator and I realised you just talk shit.
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I fed my Dog the American Dream
Well, he rolled over and he started to scream
He said, I dig the taste of salt but it don't keep me alive yeah, yeah
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