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All of those are completely consistent with markets being random! It just means that simple random models (like gbm) do not sufficiently capture observed market behaviour.
Mandelbrot proposed a fractal model of markets based on taking a cartoon stock price chart and randomly replicating it on smaller and smaller scales (like how you can build any other fractal, but with random instead of deterministic replications).
This market model has fat tails, heteroskedasticity and long-range dependence, but it is still largely random! Mandelbrot did *not* show that markets are not random. He offered an alternative kind of randomness that matched observed data better than gbm.
The fractal model of markets never caught on because it was too hard to make it practical for the kind of thing that practitioners cared about — i.e. the pricing and hedging of large derivatives books. We developed more practical models that solved the problem in a different way,
largely local volatility and later stochastic volatility models which explicitly build in time-varying volatility and hence generate fat tails and heteroskedasticity. Like all models they are wrong, but unlike the fractal model, they are useful.
That’s all I have to say about it really. Mandelbrot was not wrong, in fact he was ahead of his time — but his research led in an unproductive direction. The fetishisation of his ideas that has cropped up in the last 10-15 years is weird and I attribute it mostly to people who
have read “fooled by randomness” or “the black swan” once and largely failed to understand them 
I feel like @EmanuelDerman would have a much better take on this than I do!
Mandelbrot using a fractal to explain market phenomenon reminds me of a phrase I learned the hard way: “when your favorite tool is a hammer, everything looks like a nail.” He was brilliant, and the model is great, but it’s far from perfect (as are all models).
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