After installing SharePoint Designer 2013 and MS Office 2013 on my Win7 64-bit laptop, SharePoint Designer 2010 crashes every time I try to open a site. Can SharePoint Designer 2010 and 2013 coexist?
Yes, you can install both versions of SharePoint Designer on the same machine. They just need to be the same edition (32-bit or 64-bit).
You might be able to fix your current situation by uninstalling both. Then install designer 2013 first and then designer 2010. And make sure that both are either 32-bit or 64-bit versions of designer (also needs to be the same edition of Office 2013).