This is normal, and even expected, especially for conferences.
Because of the power imbalance, it seems as though the author might feel compelled to agree to review the paper even if they didn't otherwise want to.
Few people really want to review papers; doing it well is a lot of work for relatively little direct payoff (esp. compared to just reading a published paper). However, it's understood to be an important part of participating in the scholarly community, and the golden rule (classic definition) applies. By submitting a paper to the journal, an author is clearly expressing a desire to be part of whatever very specific scholarly community that journal serves. It is quite reasonable for an editor to request that a submitting author become a reviewer, if the editor believes they could do a reasonably good job of the review. (If the author's submission was crackpot nonsense or way off topic, they may not be a good reviewer candidate.)
Or maybe they will feel an instinct to say the paper is terrible if they believe fewer other papers being accepted might increase their own chances.
If they want to be persuasive, they will need to say specifically why the paper is "terrible" and if you reject what they are reviewing it should be because those are valid points. Similarly, if you as an editor reject what that reviewer authored, it should be because of valid shortcomings pointed out in the reviews of that paper, and a completely independent decision made without considering whether they agreed to review other papers.
Authors submitting to or published in a particular journal have an incentive to set the bar for that publication to a high level (so as to enhance the reputation of their own work, by association). Some authors may wish to qualify this, setting the bar high but not so high that their own work can't get in. Ultimately, it's up to you as the editor to decide where the quality bar is for that venue, and the reviews are just input to guide you.
I am an editor myself and I never do this on principle.
As long as the venue does not have any specific policy against doing this, I would have no qualms about it, as long as I know my editorial decisions are separated from authors' responses to review requests, and that the content-based reasons for the editorial decisions are clearly expressed in the decision letter.
As an author it has happened to me on a number of occasions, and I'm not sure how justified I am in finding it so annoying.
As an author, I don't find any extra annoyance from a review request because it comes from a venue where I've submitted. The opposite is actually more likely, especially when I'm a new submitter to a venue that I believe to be high-quality: I take it as a step of being welcomed into that scholarly community as a qualified peer.
I'm also considerably more likely to accept review requests from venues where I've submitted, because I've already reached the decision that this is a venue worth supporting. I'm also not nearly as likely to lump such a request in with the academic spam clogging my inbox with requests to review or edit for what I assume are generally low-quality venues looking for good names they can use to build their reputation and/or further a scam.
Bonus question: Is asking for a "quick opinion" (rather than a full review) in this context more or less appropriate than asking for a full referee report?
It depends: is that a normal step/activity in the venue?
You might get better full referee reports from reviewers who also have work submitted at that journal, and/or who recently received good-quality reviews (positive or negative) from a submission to that journal; such reviewers may be more cognizant of the golden rule when doing their review.
As an editor, I'd be more inclined toward requesting full referee reports where possible and appropriate, only asking for the "quick opinion" reports if I were needing more of those reports for a step in the venue's process, and lacking in reviewer candidates likely to give those but unlikely to do full referee reports.
