File copy: Add new files, but do not replace files, to a set of folders
The app FolderSync solves the problem. You can choose whether to overwrite target files or not. (Even if you choose not to overwrite individual files, you can still include new files in subfolders into your backup, if you want.)

You can also choose whether to only backup/sync modified files (date or MD5 options). Under this option, a file deleted in the cloud, will not be uploaded again, unless modified.

The app has many options e.g.


synch both directions or just f.ex to (or from) the cloud
include/exclude subfolders
include/exclude filenames containing user defined strings
synch deletions is optional (files deleted on the phone will then also be deleted in the cloud)
and many more


It is able to handle thousands of files and also works under a less than optimal Internet connection. It connects to a multitude of clouds. It does scheduling and works discretely in the background.

So far it has solved my copy needs for the cloud.

...

Another good app is ES File Explorer. It is a file manager (not a sync program). It also handles the common clouds (I think you need "Pro" version for cloud Mega.)

ES File Explorer works well. But if you choose "do not overwrite files" you can not include subfolders. (Files and folders are treated in the same way with respect to overwrite/skip.) At least I couldn't make it work. For me it also had problems communicating with Microsoft cloud OneDrive for large backups. Maybe because my Internet connection is somewhat poor. "ES" functions well with other clouds over my Internet connection,  though.