Extracting $ x $ given $ g^x, g^{x + y}, y $ Given a cyclic group $ G $ of prime order $ q $ with generator $ g $ in which computing the discrete logarithm is hard. 

Is it (still) hard to find the discrete logarithm $ dlog_g(g^x) = x $ if the 
values $ g, g^x, g^{x + y} $, $ y $ are known?
Ok, I gave the answer in my comment; however so that you can accept an answer (and so close this question out), I'll repeat my answer here.

Yes, it is still hard to find the discrete log, given $g, g^x, g^{x+y}, y$.  The reasoning is simple; if that were an easy problem, that is, if we had an oracle that, given $g, g^x, g^{x+y}, y$, recover $x$, we could then solve the discrete log problem.  Here's how we would do that: given $g, g^x$, we would select a random $y$, and compute $g^y \cdot g^x = g^{x+y}$.  Then, as we had $g, g^x, g^{x+y}, y$, we could hand those values to the oracle, which would give us $x$, solving the original DLog problem.

You also have an obvious reduction in the other direction; hence your problem is precisely equivalent to the standard DLog problem.