Why do you want to close it off? most of the option you are talking about wont reduce sound, and I would be you would get sick of drawing them shut..!

I can’t tell how much space there is between the sofa and stairs. Can you put the TV where the sofa is now and float the sofa so that the back of it creates a bit of an entrance/ foyer space? Ideally a sofa table would fit behind it to have a landing zone and to not have to just stare at the back of the sofa.

see if SJ McCarthy can come in and comment. She is the flooring guru and I believe she is in Can. look her up, send her a message, and have give her a link to this dilemma. She'll pop over. I'm sure the two 'installers' you spoke to are not flooring refinishing experts. They are probably just flooring installers. Big difference. It's not cheaper. It's costs more to tear out and buy all new. And of course that's what they told you because they're the ones taking your money! You need someone who refinishes wood floors. nothing else. Call up a wood flooring refinisher. call in 2 or 3. Talk to someone who knows what they're doing, not some random flooring guy. new flooring. $5-$10 sq ft for wood flooring, solid or engineered. (yes there is cheaper, and it's crap) now add on about $4 a sq ft to install. don't forget new baseboards. demo all of the old stuff, prob $4 a sq ft. do the math. now, match up new flooring,,$5-7 sq ft for the new wood to lace in, few more bucks for labor, plus a few thousand to completely sand and stain everything at once. (I don't know what your sq footage is) it's cheaper to leave what's there, and pay to have it matched and redone. And, you get to keep your original floors and have them look brand new.

IMO remove the cabinet by the stroller and install anice big armoire for coats shoes etc. As for anything else the kids take it to their rooms ours did it for years in a samll house .You have a ton of space right there for all kinds of storage .