Yes, thumbnails only work if you install Sumatra (and select thumbnail creation during install step using ‘Options’ button).
In Windows, generating thumbnails works by registering a dll that Windows will call to generate a thumbnail for a given file type.
Only installable version of Sumatra provides this dll, for 2 reasons.
1. Portable version ships as a single executable, for convenience. There’s no room for shipping a dll.
2. The idea behind portable apps is that it doesn’t leave a trace on your system. You can fully remove it by removing the executable file.
To register thumbnail provider we need to modify registry and we can’t allow for the dll to be just deleted by the user without first unregistering it with Windows.
As to changing the icon: there are many programs that allow editing resources inside a Windows program and an icon is just one kind of a resource.
You could create a new icon and use one of those programs to replace the icon inside SumatraPDF.exe.