Can this be used to refer to a stance on an issue, like in English, "Where do they stand on poverty?", etc?
I don't know that I've ever heard it used that way but it would probably be understood in context. Afterall, standing up to someone is expressed with the compound word "kontraŭstari". However it would probably be more likely to hear "Kion vi opinias pri malriĉeco?" or "Kiel ni solvu malriĉecon?" or even "Kian politikon vi subtenas rilate al malriĉeco?"
The question I had just before this was "Kien Sofia iras?" Why is this question not "Kien ili staras?"
The -as suffix is the present tense for verbs. The -i suffix is the infinitive. "Kie ili staras" means "Where are they standing" or "Where do they stand". An infinitive verb such as "stari" is usually subordinate to another verb, such as "Mi volas stari" (I want to stand).
For now you probably don't need to worry about the other uses of the infinitive but here are a couple examples: "Esti aŭ ne esti? Jen la demando" (To be or not to be? That is the question). "Ami estas vivi" (To love is to live). "Manĝi estas bone" (To eat is good).
"Jen mi estas, jen mi restas! " - House Atreides was more my association. Here "mi" makes sense as it's Leto Atreides's declaration upon arriving on Arrakis. His descendants merely take the same stance.
In this House Mormant quote, whoever they are, it sounds rather weak: "Here I stand"? Is there a context, or should it really be: "Ĉi tie ni staras." (Here we stand.)