1) How do I get that change in my VSC so I can push to Github?
3 retrieves: CustomApplication, StaticResource, ContentAsset
sfdx force:source:retrieve -m CustomApplication
sfdx force:source:retrieve -m StaticResource
sfdx force:source:retrieve -m ContentAsset
You can pinpoint specific files within these folders, but this was easiest for me.
There were a few more nuances in my case because I updated existing files in the Packaging Org. Next time, I'd go to each of these 3 Folders in VSC, copy an existing file and it's meta file, and then tweak to my needs. As David mentioned, this is how I'd keep Github the Source of Truth, instead of making changes directly in the Packaging Org.
2) I clicked Upload to create a new version of our Managed App. When I installed to a Dev Org, the old logo still showed. Why?
David was correct. As the documentation says, upgrading the Managed Package to new Version does not update the Logo (Setup > App Manager)