I used to work in a computer repair shop. Unfortunately this is a lot more common than you might think, and it's almost always due to poor quality plastic and nothing to do with the end user. Sadly I've yet to see a case where this is covered under the warranty either, but it's worth a shot if you want to go through the headache of their customer support.
You can try epoxy like someone else mention, but it comes with a few problems you might encounter. First you can't put too much epoxy or it'll spill over onto the computer components (which could damage them) and as a result it may not hold. Second, even if it does hold you're putting it over screws, which means you most likely won't be able to remove those screws down the road if you ever have to repair or upgrade it.
You could also just leave the laptop permanently open and never touch the hinge. Some people already this anyways with their devices so for them it's not worth the hassle. As long as nothing else is damaged and you're careful with it, this can work too.
Ultimately, the best solution is to just replace the plastic casing part where it ripped out of, which is usually either the housing around the keyboard and track-pad, or the one behind the LCD. Sadly because all the parts are likely made with poor quality plastic this is likely to happen again down the road. In your case it looks like the former, the housing around the keyboard. For some devices the keyboard isn't easily removable from this housing, so it's probably going to require you replace the keyboard as well, unless you like dealing with tiny rivets.
Oh, and you probably don't need to replace the hinge itself. like I said it's usually the brittle plastic in the housing that's at fault, not the hinge seizing and refusing to move.
Sounds like it's an issue with your router configuration and NOT your steam deck. If you have tethering on your cell phone you can use it temporarily to test if you get Wi-Fi that way, though I wouldn't download anything while tethered to your phone if you have a data cap for your cell phone.
Since it sounds like your ISP has complete control of your router I would call their customer support and let them know you have a device that cannot connect. Most likely they'll either do something like a factory reset on the router or ask for specific hardware information (like the MAC address) about the device you're having trouble with (in this case your steam deck) so they can check the setting and ensure it isn't blacklisted or something.
If they for some reason still won't help or can't get it fixed, you can always go out and buy your own router and hook it up. You'll still have to contact your ISP again to get it setup but at least you'll both have full control over the router configuration, and you won't have to pay a monthly fee to rent a router from them.
EDIT: You may also be able to factory reset the router yourself. They usually have a pin hole in the back or something you can press with a paperclip or needle that will factory reset them when held down for a few seconds. Keep in mind anyone else connected to the router will have to redo all connections for all of their devices. If you live with other people it would probably be a good idea to let them know you want to try this before doing it so they aren't thrown off the internet unexpectedly.
EDIT2: The reason you may have had issues launching games and factory resetting the device may be due to the fact that it had a Wi-Fi connection setup but it couldn't get internet. Usually when the deck is online (even if you are playing an offline game) it will still try to contact steam servers to do things like notify friends you're playing x game and update cloud saves, as well as ensure you aren't playing games on more than one device (this last one is annoying). As a result, the game wouldn't launch because it got stuck in limbo trying to reach the steam server, thinking it could because it saw it had internet access, but never getting a response because it didn't actually have internet access. Going into the settings and setting the console to offline mode would fix this issue, assuming this was the cause. The factory reset could have been something similar, it may have been trying to check online for any update during the reset. Unfortunately both of these are conjecture and not something I can say for certain, but it at least means there's a chance nothing is wrong with your steam deck. I would try getting it connected to the internet again and testing things before reaching out to Valve for support on the steam deck, since it sounds like it is mostly an issue with your router that you are unable to connect to the Wi-Fi currently.