Yes you will need to replace the battery. Since you are able to use your laptop when it is plugged-in, it is a great indicator that your charger adapter is not the issue. Make absolutely sure your computer is turned off. Remove the back cover of the laptop by unscrewing the screws and gently lifting the cover up. Locate the battery which is usually a relatively large rectangular shape device and replace it with a known working one. If this does not work, there might be an issue with the actual wires inside the laptop. If that is the case you will need to either send it back or see if a local electronic store can do something with it.

Yes you can replace the screen but it will take delicate steps for separating the screen from the touch pad behind it. Refer to this troubleshooting guide on how to do this: HP Envy 15-u001dx Screen Replacement

If your laptop is constantly running, dust can accumulate and cause your laptop to overheat and lock up. Place a piece of cloth or towel to lay the laptop down on, or some place where you won't mind dust spreading. Make sure that the laptop is completely shut off. Just need to simply remove the back cover of the laptop. Locate the fan and use a can of compressed air on the fan to get all the dust and debris off of it. Be sure to not spray to strong with the can as this can damage the fan. Hope this helps.