The short and simple answer is “Yes”. All problem above can be solve.
Initially, when you deleting a file, it will be stored in Trash, which is present on your Mac system at the right-hand end of the Dock.
The trash is only a place which shows you the files you deleted from the drive. This deleted file is just being shown on the trash and reminds you just deleted it. Here you can easily restore the deleted file at any time until you empty the trash bin. Once the Mac Trash Bin get emptied, it is impossible for you to find the emptied files again.
There are 2 ways recover files from trash mac for free by yourself:
1. The screenshot shows how to "undelete" a file. It will be moved back.
If you simply put files in the Trash you can restore them by opening the Trash (left-click on the Trash icon) and drag the files from the Trash to your Desktop or other desired location.
2. "Time machine" is a built-in backup tool that works with your Mac and an external drive, As the problem you described, I read it. The files deleted in Trash bin could be a different problem which Time machine can't solve. You can read this article and learn more. I hope it will help.
If you emptied the trash, the deleted file is still being recovered because you delete a file, the file itself still remains on the drive, the thing you “deleted” is just the space of it. The computer operating system is just free up the space of this file so that the new file can be put on the drive.
So remember this, after file deletion is happened, don’t put any new files on the drive which the lost files were stored before ’cause this will lead to data overwritten situation. Once this happens, you will lose the chance for lost file rescue.
If you stop using the drive it's possible to recover deleted files that have not been overwritten by using recovery software such as uFlysoft.www.uflysoft.com/data-recovery-mac/recover-deleted-files-fro