Can I somehow "share a folder" in Kali Live so it is accessible in Windows?
Share folder between Kali Live and Windows
You can run your Kali Live from VirtualBox or VMware. Detailed official documentation can be found here.
Then you should install guest additions in running Kali Live. Official documentation here.
And finally you will be able to create Share folder between Kali and Windows. Previous link to documentation also contains details about this setup.
samba on Kali Live
You can to do this by installing
samba server package and configure it.
There are more Kali-specific examples you can get by googling. How to Install and Configure Samba Server on Kali Linux one of them.
The most detailed documentation you can get from Samba project documentation.