What do I need to fix a defective iPhone 5S 32GB Logic Board?

Hello, trying to get my old iphone 5s, 32GB, Space Gray back to operational. It functioned just fine last time I turned it on (2-3 years ago now). There’s no damage to the screen, case, etc.

It slowly stopped holding a charge and would only stay on when plugged in. We used it more as an ipod for music (spotify, pandora, etc). Wanting to get it working again for my daughter to use in the same capacity for music apps. Also so she can text (iMessage) or face time when on Wifi from her own iCloud login (no cellular setup). Just basic functionality until she’s old enough to have a real phone. The original provider was AT&T. I’ve read that some folks unlock their phones, but guessing we don’t need to worry about that since we’re not actually trying to set it up with a phone number or cellular service? I’m also OK with losing all data that’s on it (probably already lost since the Logic Board is dead). It seems we just need a new (used-working) logic board and maybe a new battery to get her setup on the phone with her own iCloud act with basic apps and wifi functionality.

The Apple Store told me replacing the battery wasn’t the issue, the Logic Board is defective. They would charge $280 to replace the logic board. Obviously, that’s not reasonable so I started looking online. It seems there are logic board replacements available here through iFixit, Amazon, Ebay, etc.

My questions are:

1. do I need a logic board WITH the paired home button?
2. does the color matter at all on a replacement? I noticed some being solde state from/for black, gold, etc. But for an internal part, what difference does the color make - is there another compatibility issue there?
3. What else do I need to ensure the logic board replacement includes to properly match it with my phone it will be used with?
4. Will apps still function even though apple stopped supporting the iOs version this phone is capable of running on?
5. Assuming everything above checks out, guessing I should get a new battery while at it?

I can get a battery for around $15. I have seen logic boards w/o a paired home button for $11 - $24. $20-$29 w/ paired home button. So for at or under $40, I should be able to get her a working, handheld device that’s more user friendly than her ipad for taking to her friends, visiting grandparents, etc.

Please advise at your earliest. Thank you!