XBOX 360- Can I install a game from a disc that gives a Unrecognized Disc error?

So my Minecraft: Xbox 3060 Edition disc stopped working about a year or so ago, and would give me an "unrecognized disc" error. I tried my best to fix it but i failed and soon just forgot about it. I came back to it today and found out that I can install a game to the xbox itself from a disc and be able to play it without having to use the disc, and I'm wondering if there's any way to do that with my broken game.
(warranty for the game has ended a long while ago so that isn't an option.)

I need clarification on Xbox One online gaming, how does it work?

I recently bought a used Xbox One for my kids who want to play Overwatch and Minecraft. Now we currently have a Wii U system and they can play Minecraft online without anything other than our internet connection. We just recently bought the Minecraft game for the Xbox One and it tells us to use it online we must pay for XBOX Live Gold, why? The console is already hooked up to my internet connection so it's almost like XBOX wants to charge me to use my own internet connection. What type of software is in an XBOX One and why isn't my local internet connection good enough to play through with online gaming by itself? I feel like I'm being scammed. I don't need any of the other Gold benefits just want kids to be able to play online with other players through games like Minecraft and Overwatch. If I bought the PC Overwatch edition I wouldn't have to buy an extra paid subscription so why does XBOX make you do it?

My son has a WiiU console and is looking to change to a Xbox or ps?

He will be 10 this year and is unsure which one to go for.
We're torn between picking up the obvious bargains with cheap 360's & ps3's as there are plenty on the second hand market with lots of cheap games still about or just grabbing a ps4 or xbox1 as they are newer.
I personally have never played on an xbox and its been plenty of years since i last played on my ps2.
He would like minecraft, fifa and of course any lego orientated games but has expressed interest in 'hello neighbour' & 'fortnite' which of course are only available on the newer consoles.
Can anyone offer any advice as to help with our decision?