Is Mars' mantle homogeneous or heterogeneous and what might this say about mantle convection?
The mantle of Mars is heterogeneous. While there are many things that are unknown about the Martian mantle and the entire internal structure of Mars, this is not one of them. The source of this finding comes from the analysis of meteorites originating from Mars.

Analysis shows that 4.5 billion years ago - when Mars was forming along with the rest of the solar system - two distinct "reservoirs" formed in the mantle - the "Enriched Reservoir" (ER) and the "Depleted Reservoir" (DR). The ER had high ratios of lanthanum/ytterbium and $^{87}$Sr/$^{86}$Sr, and a low ratio of samarium/neodymium. The DR had the opposite ratios.

Eventually, meteorites known as shergottites formed, mixtures of both of these reservoirs. A graph (which I'd insert here, but I'm unsure of the copyright on it) entitled "Heterogeneous Martian Mantle" illustrates how these meteorites have a wide range of composition compared to rocks from Earth. This provides compelling evidence for a heterogeneous Martian mantle.

At the moment, I can't find enough information to answer the follow-up part, though the convection certainly did stop.



Sources:

PSRD article 1

PSRD article 2