The First Problem
After I installed Cisco Webex Teams, Skype began to crash each time I launched it.  I found some articles stating you can only have one application use Outlook for status updates and if you have both Webex Teams and Skype, it can cause Skype to crash (see https://collaborationhelp.cisco.com/article/en-us/gk4yog and https://collaborationhelp.cisco.com/article/en-us/yf1gc7).  My guess is this is what was causing Skype to crash.  However, it sounds like this issue didn't occur with others in our company so what was different with my install?  Well, my install was a little unique as I apparently already had a "personal" Webex account from when we used Webex years ago for webinars within our company.  With a personal account, you have options to integrate with Microsoft Outlook – my guess is one of these settings was enabled prior to "converting" to an enterprise account (with enterprise, these settings are not available).
 
The Solution to the First Problem
  1. First, I uninstalled Cisco Webex Teams.  After uninstalling Webex Teams, Skype would work fine, but if I reinstalled Webex Teams, Skype would crash again.
  2. Second, I tried installing Webex Teams and then running a "repair" on Outlook.  This seemed to fix the issue with Webex Teams and Skype (I could have both installed and Skype would not crash); however, this created my second problem – Outlook no longer showed "online status" or "presence" for company users (see internal staff status when I send emails in the "To" field).
The Second Problem
After running the repair on Outlook to fix the problem with Skype crashing when Webex Teams was installed, Outlook no longer display the "online status" or "presence" – while this doesn't seem like a critical issues, it has helped me ensure I don't send internal emails to customers with similar names in the past, so I wanted to get it fixed.
 
The Solution to the Second Problem
  1. First, I found the setting where you can enable online status (https://support.office.com/en-us/article/use-skype-with-outlook-to-display-a-contact-s-presence-information-b1509222-2c5d-4cd4-bff7-508d2b6f410d) but it was checked.
  2. Second, I researched the Registry settings for Skype and Cisco Webex Teams and found the following two settings I needed to change in order for Outlook to show "online status"
    1. Computer\HKEY_CURRENT_USER\Software\IM Provider
      1. Had to change the DefaultIMApp from "Cisco Spark" to "Lync"
    2. Computer\HKEY_CURRENT_USER\Software\IM Providers\Cisco Spark
      1. Had to change "UpAndRunning" from 2 to 0