How much do airlines care about stage checks in a part 141 university program? Also how much do they care about part 91 failures? Like private, instrument, etc?

You can't "fail" a stage check. You can "incomplete" one.

(Totally unfair, but this is why 141 makes everyone look like a rock star.)

Also interested in this because I am trying to figure out if I should answer yes this question on an app.

Have you ever received an unsatisfactory evaluation during any pilot training, testing, or checking event?

Just be honest and up front. Did you learn from the "failure" on the stage check?

to be clear failing a 141 stage check is never a failed check ride. Even if it's the final check, its still not a failed check ride. If the application is specifically asking about stage check then of course don't lie. like jcountry said it does artificially make everyone look like a rock start.

No, you can fail a stage check.

Yes you can earn an "unsatisfactory" grade on a 141 stage check, an incomplete is when you or the designated examiner need to discontinue due to illness, time, etc. A completion of an FAA approved 141 course replaces taking a checkride. Your "checkride" is just the last lesson of that course. Often the school will have a stage check right before the final lesson because they need to maintain a %80 pass rate of that final lesson if it pertains to a certificate or rating in order to keep their 141 certificate.

Airlines are aware of how it works and often ask if you have unsat any STAGE checks. Just be honest, I don't know of anyone that made it through a 141 program without any hiccups, they know it's difficult; especially early on in your training.

You don't get a pink slip but a careful eye can spot it in a log book. Just answer the question as asked. If they specifically ask about stage checks or end of course rides. Then be honest. If they ask about FAA busted rides then be honest about that