Hello Chris,
Thank you for visiting Microsoft Community and providing us with the detailed description about the issue.
As per the description, I understand that you are experiencing issue reinstalling Twitter and getting error code: 0x80050007 on the system.
I certainly understand your concern and will try my best to help you.
I really appreciate all your efforts in trying to fix the issue.
I would suggest you to refer to the suggestions provided by “John Rubdy” replied on February 11, 2014 in the following Microsoft Community link given below and check if that helps.
**Note:**The Modern app troubleshooter is a built in troubleshooter that can detect and fix most of the app errors automatically. So, I would suggest you to download and run the Modern app troubleshooter to fix any app errors.
Note: The Windows Store will automatically notify you if there’s a problem with an app and try to fix the problem. If you continue experiencing problems with an app, or if you're unable to open the Store, try running the Apps troubleshooter. This tool can automatically identify and fix some problems with apps and the Store. It's only available in English, but will work on PCs running any language.
Note: Antivirus software can help protect your computer against viruses and other security threats. In most cases, you shouldn't disable your antivirus software. If you have to temporarily disable it to install other software, you should re-enable it as soon as you're done. If you're connected to the Internet or a network while your antivirus software is disabled, your computer is vulnerable to attacks.
For reference:
I would suggest you to refer to the article given below and check if that helps.
Troubleshooting problems with an app
http://windows.microsoft.com/en-GB/windows-8/what-troubleshoot-problems-app
I hope this information is helpful.
Please do let us know if you need any further assistance, we will be glad to assist you.
Thank you.
Jayant