If drive encryption is done at the folder-level, do all the files inside the folder get encrypted as well?

Yes, the files inside the folders will be encrypted. All the files will have the same password that was used for encrypting the folder.

If drive encryption is done at the folder-level, will it also encrypt the files in the sub-folders too, if there are any?

Yes, they will also be encrypted.

If I add a new file into an already encrypted folder, will the new file get automatically encrypted?

Yes, it will get automatically encrypted with the password of the folder-level encryption.

How does backup and restore work for encrypted files? 
Everytime a file is encrypted-decrypted, its FileID changes. Hence, each of the file will be backed up because the fileID's are different.


How does encryption work for Team Drive? Specifically can users encrypted at the folder level in Team Drive?

You cannot apply encryption for a Team Drive, the option is not even displayed when right-clicked. But for the files and folders under the TeamDrive, encryption works the same way as Drive. 

I write custom scripts in his sheets. Apparently, each time he decrypts the sheets the script is deleted. Is this as per design?

This is a technical limitation from Google. So, scripts in the sheets will be lost when it is encrypted-decrypted.

Second, when I am decrypting the file on my account, it also is decrypting on my clients accounts.
This is as per the functionality. Because  the same file is shared to client too. 

Is the pass word decryption a one time thing, or will the file automatically re encrypt on its own?

At the time of encryption, we have option to select auto encryption. If customer checks the auto-encryption checkbox, it will automatically re-encrypt the document after 15 to 20 min. Please refer THIS LINK for more information.