How Good is The Evidence For Snowball Earth?
Paul Hoffman's argument for Snowball Earth

So, first off, I'm just a guy with access to JSTOR and an interest in geology, so perhaps I'm not really qualified to assess "how good" the evidence is. But, I can read through the papers and summarize some stuff for you, which perhaps is sufficient to answer the question. 

So, as suggested in the title, serious consideration of the Snowball Earth theory started with a Paul Hoffman paper in 1998, free copy linked here. The evidence in the initial paper is thus:


Inorganic $\delta^{13}$C showed decreases during glacial periods between 750 and 550 Ma that were 'enormous in comparison' with variation seen at other times in the Earth history. This references previous stratigraphic work by Alan Kaufman, a co-author on the paper we are discussing.
Based on a theory by Kirschvink (the geologist who proposed the Snowball Earth theory), an ocean completely sealed by sea ice would quickly become anoxic, leading to chemical reactions that would cause the observed changes in carbon deposition.
New stratigraphic work in the Otavi Group formation in Namibia confirms the drop in inorganic carbon, as well as providing evidence of glaciation on land at a latitude of ~12 S (latitude determined by paleomagnetic data; age determined by radiometry). 
The geological evidence suggests that the proportion of organic carbon in total carbon burial drops to "virtually zero" immediately after the glaciation begins. 
The paper then presents evidence that a large volcanic event with C)$_2$ levels > 300 times the present could "undo" the Snowball Earth event. 


In the conclusion, the authors say that a Snowball Earth explains the evidence from carbon isotope excursions, but no alternative hypothesis does. This is discussed more in detail in Hoffman, et al., 1998a (an article by the same authors, the same year) and Hoffman and Schrag, 2002. Among hypotheses rejected include a period of ocean stagnation and overturn; the Earth having a larger axial title until the end of the Proterozoic, true polar wander, a major bolide impact, deep ocean anoxia and others. 

The point I want to make here is that the questions that you ask in the OP don't fit into the original explanation for why there might have been a Snowball Earth episode. It is true, that without accurate fossil dating, radiometric methods can't prove that near-equatorial glaciers occured in multiple places at the same time. But, that isn't really a required proof. Evidence for Snowball Earth is "good" as long as it is the most likely explanation for the observed paleomagnetic, stratigraphic, and carbon isotope data. 

That last is the part that I am perhaps not qualified to speak on; you'd have to do a full literature review on both Snowball Earth and alternate theories to determine which has the best evidence supporting it. But, clearly there are many geologists who thing that Snowball Earth is the "best" hypothesis. 

Answers to specific questions

Although at the beginning of this period, there were, so far as we know, no advanced invertebrates able to leave easily identifiable fossils, there were by the end of it, so how were they able to thrive and evolve under the ice?

From Kirshvink, et al., 2000, 


  Finally, we speculate that the predicted severity of climate and
  geochemical change during the snowball event and its aftermath may
  have forced enormous biological adaptation and left recognizable
  signatures in the phylogenies of extant organisms and their genes.


The extreme geochemical conditions include significant changes oxygen and iron availability. Thought the above article specifically addresses an earlier Snowball Earth episode, the suggestion is that the few organisms that did survive the glaciation had novel geochemical adaptations that allowed them to explode into the much more friendly world environment that existed after glaciation. 

Indeed, the Ediacara fauna appeared after the glaciations had passed; from Hofmann, et al., 1990 (not the same Hoffman/Hofmann; also couldn't find a free full copy):


  The Ediacara fauna generally regarded as containing the oldest
  metazoan fossils, has been described from about 25 localities around
  the world; all occurances to date place it above the highest
  Proterozoic tillites of the Varanger glacial epoch.


The Varanger glacial epoch is what is now theorized to be the Cryogenian period, divided into two glaciations, the Sturtian and Marinoan. 

So, overall, the explosion of evolution and diversity happened after the glaciation passed, and it happened because radial new biochemical processes had to be evolved to survive in the radically divergent marine chemistry of the frozen ocean. 

There must also have been early marine plants under the ice (there were none on land), so how did they survive?

Eukaryotic algae (like red and brown algae) existed before the Cryogenian, and survived until after. There are many mechanisms for them to have survived. Firstly, algae do live on the snow on the terrestrial surface of Antarctica; such "snow algae" could have survived on Snowball Earth. 

Secondly, there is no consensus that the surface of the Ocean was uniformly thick. Some modeling, for example from Pollard and Kasting, 2005 indicates that the sea ice was on the order of meters thick in the equatorial regions. In that case, algae could have easily survived on the underside of ice sheets, as it does in the modern arctic and antarctic. A thin, cracking, and periodically melting (under tropical solar radiation) state would be consistent with conditions in the polar regions today, and is quite capable of supporting life. 

Couldn't there be other explanations for these apparent glacier tracks, apart from Snowball Earth? For example, Australia may have been closer to the S.Pole then, and some of the locations where evidence of glaciers was found might have been much higher above sea level than they are today.

I haven't done a deep dive, but the latitude of various points of the Earth's surface can be estimated by paleomagnetic data. Note that the many of the alternate explanations (such as increased axial tilt and true polar wander) also include mechanisms for explaining why the magnetic record is wrong (i.e. the magnetic poles were not where we though they were). 

However, this is something of an Occam's Razor situation. In order to explain tropical glaciers without Snowball Earth, you have to explain how the Earth was physically rotating on a different axis. While possible, it takes a lot of energy to make the Earth move from one axis of rotation to another. Snowball Earth is a simpler explanation from the energy physics point of view. 

As mentioned earlier, there are two links to papers where Hoffman and company dismiss various alternate explanations.