Do solar winds / mass ejections produce thrust on sun? For every action there's a contrary reaction, so can those mass ejections make the sun move?
Of course, the mass of the sun is immensely bigger than what's being eject but even being trillions of times smaller so can those ejections thrust in a trillionth of the velocity that they get?
 A: 
Of course, the mass of the sun is immensely bigger than what's being eject but even being trillions of times smaller so can those ejections thrust in a trillionth of the velocity that they get?

Yes, of course this is true.  Both the solar wind and coronal mass ejection (CME) can impart net impulses to the Sun.  The former must be non-uniformly distributed in order to do so, which it often is.  The Sun has a mass of 1030 kg while a typical CME has only 1012 kg, i.e., ~18 orders of magnitude smaller.  So if we assume the CME "pushes off" the Sun and the Sun acts like a rigid body (i.e., elastic), then from conservation of momentum the change in the Sun's speed would be ~18 orders of magnitude smaller than the CMEs.  A typical CME leaves the sun at, say, ~500 km/s.  That would correspond to a speed change on the Sun of ~0.5 pm/s (picometers per second or ~5 x 10-13 m/s).
The Sun is not a rigid body, so the recoil is more of a local compression with dissipation losses, i.e., inelastic.  There are also magnetic field topology changes and reactions, which also affect the net recoil.
A more important issue, perhaps, is the loss of angular momentum.  The launch of a CME (and the solar wind) generates a net torque on the Sun, which accumulated over very long periods of time, results in a decrease in the solar rotation rate.
A: Yes they do, because as you mentioned, to every action there's an equal and opposite reaction. However, because the solar wind is in all directions, it produces no net thrust on the Sun.
