Well, there are a couple of things to consider: (1) How’s the air flow to the unit? (2) The internal power supply might be bad, and need to be replaced (3) Have you tried using compressed air to blow out any dust / dirt? (4) IDK if applicable, but if the unit has been hit really hard or dropped, something could have come loose internally
I'll take a crack at this. First, take note of how long it takes before things go wrong (write it down). Perform this action three times minimum. Also, see if there are any problems with your internet service in your area, and if so, at what times such actions occur, noting if they coincide with your dropped connection. Run diagnostics on your router. It may be the router has gone bad. It may also be that it needs a firmware update (check with the manufacturer's site). Sony seems to be a lot better about being able to troubleshoot their devices, so check to see if there are any diagnostics protocols or software that can be run. Try finding another ethernet cable and plugging it in to the RJ-45 port.
So, I've done a fair share of iPhone repairs over the years and I can tell you that when it comes to doing your own repairs, especially with Apple-brand products, you get what you pay for - case in point, a $5 replacement battery is absolute garbage, as you have seen. Now the problem is not only a crap battery, but you may also have damaged the circuitry in the phone itself, as you need to know the exact ID numbers (model, etc) of the OEM part. On top of that, with the way the screen is acting, it sounds like the battery may have become "bloated", which pushed on the screen itself, causing damage. I hope your phone isn't damaged to where it can't be fixed, but fear it may be. Your best bet, if you're still intent on fixing it yourself, make sure you understand the ins and outs of the device and that you have the correct parts and tools.
Unfortunately, you can't use Dell-brand recovery software on an HP. However, per what flogo2003 said, you should be able to get into the recovery management option via a Windows 7 recovery disc. If you know anyone with a pc that has Win 7 on it, you can have them burn off a recovery disc for you to use.