Determining linearity, causality, memory, and time invariability from a picture of the signal
Basically you can't. In basic information theory a system has an input and output and the terms you are asking about are defined in terms of the relationship between output and input. It's difficult to determine what the input to saccadic eye movements would be. It's mostly an unconscious process controlled by the neural periphery.

In order to answer your question you would first define an appropriate input signal and than look at a screen shot that shows both input and output simultaneously.

FWIW you could potentially argue that it's causal simply because it is a real physical system and all physcial systems are causal.