In gauss' law, is E necessarily attributed to the charge q?
I don't understand the question. Gauss's law relates the electric flux through a closed surface to the charge enclosed. The portion of the electric field that creates a net flux can be attributed to the net charge within the surface: if no net charge, then no net flux.
Gauss' law is a relationship between the total flux (electric field times area) through a closed surface to the charge enclosed by that surface. Part of the electric field at each point on the surface is due to the sources inside the surface while other parts of the electric field is due to sources outside the surface. Is that what you're refering to?
Separate names with a comma.