I want to throw out a thought experiment:
If you or, if you are a parent, your child gets into both Michigan and Ohio State and get’s a generous scholarship from the latter, then why choose Michigan at a higher cost? If the student gets into UM, then clearly he or she is a great student and will earn a spot in the Honors college at Ohio State.
So then, honestly, what is the allure of going to Michigan over OSU Honors? Will job recruiters look at the Michigan grad differently from the OSU Honors grad? Will law schools, med schools, or B schools look at Michigan grads differently from the OSU Honors grad? And, perhaps most importantly, will Michigan ever win another football game vs Ohio State?
To answer your first question, you would choose OSU Honors over UMich, if OSU has given you a “generous scholarship.”
**Saturday, November 30, 2019. ** 
Here’s the Leerfield Director’s Cup, where UMich finished #2 and OSU #10. There are more sports than just football at these two schools.
https://nacda.com/documents/2019/6/11//June12OverallDi.pdf?id=3669
In the order of your questions: Some. No. Unlikely.
If finances are an issue it would be a no brainer to choose OSU over Michigan.
My Ohio raised daughter did not have a good OSU visit. Felt her major was too grad focused and not enough undergrad research opportunities. She also found it too big. We also had other things going against us - we were hit up by a group of panhandlers outside the student union (not normal from what we’ve heard) and our tour guide was just not good - just didn’t seem to like the university very much and wasn’t a motivated student. She also felt it was “too close” to home and knew too many people going there. And, she wanted a more suburban/rural setting. D was on campus multiple times throughout HS and just was underwhelmed each time.
Our Michigan visit was one of the best. The engineering tour hands down was the best of any school we saw (we made 15 school visits). Our tour guide was in the same major (you could pick which guide you went with at Michigan), was doing research that was interesting to my daughter, and they did a very thorough engineering facilities and lab tour. There were some negatives for her too but it was her #2 school choice. Had she not gotten into her top choice, she would have stayed on the WL and done everything she could have to come off.
Do I think job and grad school options will be limited by going to OSU honors? No I don’t.
Note: some students in OH wouldn’t apply to Michigan if their lives depended on it. Too big of a rivalry.
Sometimes I doubt the veracity of your posts–but okay.
Are you directing your comments at me?
I’m located in Silicon Valley. I took our puppy to the dog park on Saturday. Our timing was off-hours, so very few dogs, but TWO different dog owners were wearing Michigan shirts. Locally, Wednesday nights are food truck nights. Last Wednesday, a couple of individuals were wearing UMich paraphernalia. UMich alums seem to be plentiful around here.
I’ve only known one OSU grad here. I haven’t spoken to him in about 5+ years. And I cannot remember ever seeing another OSU grad. 
That’s my informal poll based on a sample size of billions. 
Neither here nor there, OSU is great school and you’ll get a great education. But so is Penn State!
Would you really go to Michigan if you really wanted to go to Silicon Valley after college?