Will the following process $$r(t)=\int_0^ta(s,t)dW(s)$$ be adapted to the Brownian motion $W(s)$? Will $r(t)$ be an Ito process?
Edit: Maybe I should rephrase it a bit. The question is: does dependence $a(s,t)$ on $t$ (which is not the variable for intergration) interferes somehow with the usuall requirement for Ito integral that $a(s,t)$ must be adapted to the filtration generated by W?
The answer is: it does not whatsoever.
As explained in a comment, if the function $(s,t)↦a(s,t)$ is deterministic, then the process $r$ is very much adapted to the filtration generated by $W$. "Not adapted" does not mean "which depends on $t$", but "not measurable with respect to some sigma-algebra". And deterministic objects are measurable with respect to EVERY sigma-algebra.