You can copy files and folders in so many ways with FileBrowser Professional. Here are a few.
Simply tap the file's actions menu and select "Copy" or "Move". Then navigate to the folder you want to paste your files into and tap "Paste 1 File Here" in the top right of the screen.
If you want to copy files or folders to the iOS clipboard to use them in other apps, select that from the file actions menu.
If you want to copy multiple files see here.
This method is more like old FTP clients
To exit split screen mode, grab the handle in the middle of the screen and swipe to the right until it disapears.
Simply tap the file actions menu and then "Share", select "AirDrop" and choose the device you want to share the files with.
You can also share multiple files, simply select all files and use the toolbar actions menu and tap "Share" then "AirDrop".
On the other device the file automatically opens in the files app, simply tap the share icon in the menu. Then select FileBrowser Professional from the set of icons.
If you can't see FileBrowser Professional in the set of icons, scroll the icons to the far right and tap the "More" icon. Then select "Edit" and tap the plus icon next to FileBrowser Professional, then tap "Done" then "Done" again, FileBrowser Professional should now be in the list.