Yes, indeed the Q in the electric field formula is the charge that creates the electric field around the charge.
Yes, both the charged alpha particle and gold nucleus would produce a resultant field at points around them.
That resultant electric field at that point is meant for calculating the resultant interaction between gold nucleus and another charged particle at that point and between charged alpha particle and another charged particle at that point.
r1 is the separation between the alpha particle and the other charged particle,
r2 is the separation between the gold nucleus and the other charged particle,
Qα is the charge of alpha particle,
QAu is the charge of gold nucleus.
Key idea is the alpha-charged particle interacts via the electric field of the gold nucleus.