Depending on how many you ask, and how you ask them, questions are good.

It’s all about moderation and what you ask. Ask interesting things that show that you are listening and be relatable by telling your own stories. It gets inquisitive if you don’t share anything yourself

Depends on the tone of voice, if it's nice it's nice.
If they sound angry, it's inquisition.

Questions can aim at both! When I ask you about your new glasses it might be due to an interest in the subject since I'm thinking on getting new glasses myself, or it could be a socially acceptable pretext for inspecting and admiring your eyes. In order to prolong the time and indulge in doing so I might stick to asking more about the glasses which then sounds quickly like inquisition.
Conversations hardly work on a single level. There is always subtext, meta meaning and bluntness. Tell me more about that conversation?

Depends a lot on tone and context and all, and the flow of the conversation... Like if they talk about their sick dog and you ask about its name/age/breed, that's natural... Asking why you don't take it to the vet enough comes off as judgmental and accusatory

It could be both depending on what's being asked, how, when....

Totally depends on what they're asking, how they're asking, their intentions

I think people who take it as interrogation are full of themselves and think they’re more interesting than they are

Both, I'd say it depends on the context