Another workaround: Go back to the desktop and try AnyTrans ($39.99 for a single computer), a file manager for iOS devices that has an integrated downloader supporting 900 sites, including YouTube and Facebook. It'll transfer the videos to the iPhone for you over the USB cable. Even if you don't pay for AnyTrans, the download option remains and is free forever.
Backup services vary widely in how they set up and perform backups. For example, the totally hands-free Backblaze automatically encrypts and uploads all your important files without any input. On the other hand, services such as IDrive and Acronis True Image let you choose specific files you want from a file tree. Note that some services restrict you from backing up specific file types or using particular sources, such as from an external or network drive. Make sure the service you choose supports all your needs.
Bitcasa Infinite Drive is relatively new, but it's one of your favorite cloud storage providers in general, mostly because they offer virtually unlimited stroage for syncing and backups. When we say unlimited, we mean it—some of you are using terabytes of storage with Bitcasa. It's not primarily a backup service though, and while it was built for file syncing and storage, the Bitcasa desktop client does support regular file backups. Bitcasa supports Windows and OS X, and encrypts all of your files before uploading so they stay safe from prying eyes. Bitcasa even keeps revision history, so if you've backed up a file multiple times and need an older version, you can pick it out and restore it. Plus, you can use the Bitcasa mobile apps for iOS, Android, and Windows Phone to access your data on the go.