I'm in SAS Honors and will try to answer your questions to the best of my ability.
In my experience, yes. I live in the honors housing on Busch campus so I can only attest to McCormick Hall being a bit more quiet and studious generally. McCormick is comprised of suites and on each floor there's 2 pretty big study lounges. Usually you can nab a spot and it's a great place to buckle down and get work done. Brett Hall is the honors housing on College Ave (Alexander Library is a really close walk), Jameson for Cook/Douglass, and I believe the 8th floor in Lynton Towers on Livingston is reserved for honors. I'm not sure how the housing works for SEBS Honors.
They have interdisciplinary honors seminars and department-based honors courses. You need to have 4 completed by the time you graduate. I'm taking an interdisciplinary honors seminar and I already took Chemistry Honors. Sure, it sounds better but nobody really cares unless you're looking for an ego boost. You do go a lot more in depth and are challenged, if that's what you're looking for. Yes, the classes are a lot smaller and yes, you take the classes with other honors students.
Finding internship and research opportunities really depends on how proactive you are. I believe ARESTY is open to everyone but you do have to complete a Capstone project your senior year to graduate with honors.
I have SAS Honors put down on my resume. I'm sure it holds some kind of weight but how much I'm not too sure. Really, I think the best part about being in Honors is the housing. Also, even though the honors sections are a lot harder, the class sizes are significantly smaller which make it easier to interact with your professor as well as opposed to being in a 400 person lecture in which everyone blends into one another.
Hope this was somewhat helpful! Feel free to PM me with any other questions.