On what computer operating system is your Premiere Elements 13 running?
Problems with the same symptoms can (often do) have different causes. So, let us zero in on the details specific to your "Premiere Elements 13 will not open beyond Welcome Screen" issue.
The usual drill questions...
1. Does the problem exist with and without the antivirus and firewalls disabled?
2. Do you have the latest version of QuickTime installed on the computer along with Premiere Elements and are you running the program as administrator?
3. Is your video card/graphics card version up to date according to the web site of the manufacturer of the card or the manufacturer of the computer?
4. Did you install the program to the default Local Disk C location?
5. Assuming that you are using Windows 7, 8, or 8.1 64 bit, please go to
Local Disk C
Adobe Premiere Elements 13
and in the Adobe Premiere Elements 13 find the file specifically named Adobe Premiere Elements.exe. Right click that file and select and apply Run As Administrator. Then double click that file and determine if Premiere Elements will open to its workspaces.
If none of the above resolves the issue, then (assuming Windows for now)
uninstall the usual Control Panel way
do the free ccleaner run through (regular and registry cleaner parts)
reinstall with antivirus and firewalls disabled. Do not download and install the 13.1 Update until you are certain that 13 is working as 13.
Please consider the above. Based on the details in your reply, we can decide what next.