I would not have thought that the fact that you're running Windows 7 is the issue. Upgrading to 10 just to see if this will resolve is a big step - especially if it does not resolve the issue. Mind you, there are other positive reasons to upgrade to Windows 10 (fresh install of course). However, that's a discussion for another day.
I never had this issue when I was running Windows 7 64bit either. I use an external MIDI module so I would have heard notes if this issue occurred here too.
To me, this looks like a specific issue with the Midiman Midisport 4x4 itself (Are you sure you're using the latest drivers from the manufacturer's website) - even though it only manifests in CbB.
Notes should not be triggered when CbB starts. The only MIDI data that may possibly be sent on start up would be sysex data. This would be configured with the Sysex view. In fact, on your start startup template, check the sysex view (ALT+SIFT 7) to make sure no sysex banks are set to AUTO transmit. I am doubting it this though because you also say that you are getting note on data sent when you solo/mute/record/playback tracks. That is really strange and something I have never seen in the twenty years I have been using Cakewalk products.
Are you able to configure the Roland XP-80 (or any of your other devices) to show you the specific values being sent through the MIDI ports?
It also might be worth dropping a note to Bandab Support to see if they can offer some insight. https://help.cakewalk.com/hc/en-us/requests/new