If you filter out any feature observed in at least one sample, that would remove everything! Because if a feature is observed in 0 samples, well, it would not exist in your feature table (and current default behavior for the filtering commands is to drop samples/features with 0 frequency after filtering).
Great question — see this post. Feature can mean any kind of observation in a feature table... in your case I assume feature = sequence variant, but it can also mean taxa if you are using a feature table where taxa are the observations.
No clue — you should consult the cytoscape docs for more details, but maybe @michael.shaffer knows?