This has intentionally been done by Apple so that you will lose trust in independent repair. The battery itself is fine, and will work fine. https://www.ifixit.com/News/apple-is-locking-batteries-to-iphones-now
There’s no possibility of overcharging the battery as the charger disconnects from the battery when you reach 100%. Read this post on how you can care for your battery: https://www.ifixit.com/News/how-to-care-for-your-laptops-battery-so-it-lasts-longer
Just before you order a new screen, open the phone and check that all the cables are firmly attached. There’s no harm in checking and it could save you time and money. Only then should you replace the screen
There is a product called an “iMesa touch ID repair kit” that you can buy from the internet. However, this is difficult to apply and requires microsoldering skills, and it depends on where the cable broke. If you are not experienced, then Jeff Suovanen’s answer is probably the best - you will either have to give it apple or a microsoldering repair shop.
The problem with a branded housing is that it has an apple logo on it, which is copyrighted, making it illegal to manufacture. You can either buy a new unbranded housing, or a branded used but high quality housing. iFixit sells unbranded ones here iPhone 7 Blank Rear Case. (I just put that there to help others who come to this page later on). You mentioned that you needed a branded housing, and that probably won’t be of much use to you. For branded housings, I have always found good branded housings on eBay. I would try searching there
The first thing you would try would be to check if the cables are correctly in place. Open up your phone, disconnect the battery, then unplug the screen cables and reconnect them again. The next thing you would try would be a new screen. Try replacing your screen with this guide iPhone 6 LCD and Digitizer Replacement Then, if that doesn’t work, then if would be a problem with the touch IC. You mentioned that your phone was bent. Can you see any weird grey bars on the top of the screen? That is often a symptom. Having a bent phone can damage the touch IC and display ICs, which stops the touch screen from working. If a new screen doesn’t fix it, I would suspect it is a problem with these ICs. It is a common issue in the iPhone 6, Bring it to a micro-soldering repair shop, and they should be able to fix this.
With water damage, there can always be problems down the road. The first thing you should test is whether it is a problem with the screen. If you still have the broken screen, test and see if the home button works on that. If it doesn’t work on that, then the chances are that the home button is broken. Also inspect the flex cable for any tears.