Given $ g^s, g^y , g^r, g^t, g^{st-rs}, g^{(yr+d)/t}$ , is it hard to distinguish $e(g,g)^{syr}$ from a random value?
No, it's no easier than the standard DBDH problem.

Here's the reduction that shows that: suppose that we have an Oracle that solves your problem (given $g^s, g^y, g^r, g^t, g^{st-rs}, g^{(yr+d)/t}, e(g,g)^x$ is $e(g,g)^x = e(g,g)^{syr}$?)

Now, suppose we're given $g^s, g^y, g^r, e(g,g)^x$, and are asked whether $e(g,g)^x = e(g,g)^{syr}$.

What we do is select a random values $t'$ and $z$, and compute $g^{t'} / g^r = g^t$ and $(g^s)^{t'} = g^{st-rs}$; we don't know the value of $t = t' - r$, but we know that it exists).  We further compute $g^z = g^{(yr+d)/t}$ (again, we don't know the value $d = zt - yr$; but we don't have to know).

Then, with these values, along with the values we got with the original query, we ask the Oracle (and get the answer).