NTFS is considered pretty stable on Linux now. It should be safe to use indefinitely.
If you’re worried about the lack of Unix-style permissions and attributes in NTFS, then getting BTRFS or ext4 on Windows may be a good choice. Note that BTRFS is much more complicated than ext4, so ext4 may have better compatibility and lower risk of corruption. I used ext3 on Windows in 2007 and it was very reliable; ext4 today is very similar to ext3 from those days.
The absolute best compatibility would come from using a filesystem natively supported by both operating systems, developed without reverse engineering. That leaves only vfat (aka FAT32) and exfat. Both lack Unix-style permissions and attributes.
Have you imported the tails-signing.key yet? Usually you can double-click on that to import it using whatever graphical gpg frontend is set up on your system. It may ask you how well you trust the owner of the key. You can answer that question however you want without affecting this verification process.
Next, it looks like you run the instructions from this page: https://tails.net/install/expert/index.en.html#verify
Some of those command line parameters look a little paranoid. The basic command you want to run is:
gpg --verify somefile.sig somefile.img