Does it make sense to buy a compatible Mac external disk? What is the advantage of buying a compatible Mac external disk?
Using Tuxera NTFS I can have an external disk for both, the windows partition (or any windows machine) as well as for the Mac one.

Does it make sense to look for a compatible external disk nowadays? 
What would be the best way to deal with the PC/Mac compatible problems in terms of disk formatting? Is using Tuxera a nice way to deal with it?
Actually you can format any hard disk to macos journaled if you like. So buy the hdd you want, partition it and format it as you like.

If you format it as FAT you can use it on both win and mac for read/write, but you cannot use it for time machine backups (time machine requires macos filesystem).