  We study a class of McKean--Vlasov Stochastic Differential Equations
(MV-SDEs) with drifts and diffusions having super-linear growth in measure and
space -- the maps have general polynomial form but also satisfy a certain
monotonicity condition. The combination of the drift's super-linear growth in
measure (by way of a convolution) and the super-linear growth in space and
measure of the diffusion coefficient require novel technical elements in order
to obtain the main results. We establish wellposedness, propagation of chaos
(PoC), and under further assumptions on the model parameters we show an
exponential ergodicity property alongside the existence of an invariant
distribution. No differentiability or non-degeneracy conditions are required.
  Further, we present a particle system based Euler-type split-step scheme
(SSM) for the simulation of this type of MV-SDEs. The scheme attains, in
stepsize, the strong error rate $1/2$ in the non-path-space root-mean-square
error metric and we demonstrate the property of mean-square contraction. Our
results are illustrated by numerical examples including: estimation of PoC
rates across dimensions, preservation of periodic phase-space, and the
observation that taming appears to be not a suitable method unless strong
dissipativity is present.
