If you plan on getting another iPhone
iCloud backs up everything on your iPhone—every game's data, every app's data, your health data, your messages, your phone logs, your settings, your Homekit data, your photos and videos, everything. If you get a new iPhone, when setting it up, you can choose to set it up from a backup, and that new phone will have all of the data your current phone has. The sync is as complete as backing it up to a Mac. Also—try the Google Photos (not Google Drive) app for awesome photo backups.
If you need any help backing your phone up with iCloud or iTunes, check out Apple's support article.
Note: you can very easily get a new sync cable for your iPhone from Amazon. Here's a reliable one, and it's just $6.
If you plan on switching to Android or you're not sure what your next phone will be
Save your photos with Google Photos: download the app for iOS and it's really easy to back up all of your photos, and since Google Photos comes preinstalled on many Android phones (and is available from the Play Store on all others), your photos will be right there waiting for you.
Save contacts by going to iCloud.com and exporting them all as a VCard, then importing this into your Gmail contacts; see a guide here: https://support.apple.com/kb/PH3606?locale=en_US.
Notes synced to your Gmail account are just waiting out there in your google account, but notes synced to your iCloud account must be manually exported with Copy and Paste. Here's Apple's official guide on exporting your iCloud notes:
To copy notes, open the Notes app at iCloud.com. Copy the text of each
note and paste it into a document on your computer, such as a Pages or
TextEdit document. Save the document to your computer. To export your
notes as PDF, open the Notes app in OS X Mountain Lion or later.
Select the note, then click File > Export as PDF and choose a
location.
Emails are self-explanatory, just hook up the Gmail app on your next phone.
Calendars are pretty self-explanatory too, they should all be tied to accounts in the cloud you can sign in to on an Android phone.
Your phone number is hooked up to your Sim card, but be sure to deactivate iMessage if you switch to Android or you won't get some text messages! Here's a guide on that: http://www.imore.com/how-to-disable-deactivate-imessage-iphone-ipad
Anything else on your phone can likely be exported in its own way, and more than you may realize is tied to the cloud.
Doesn't look like there's a way to single out the Desktop via the 'Manage Storage' GUI.
However, you can always just symlink any directory from your iCloud drive; that way you can ensure that only the Desktop folder is being synced. This is with the caveat that while your Desktop files are being stored in iCloud, they're still taking up space on your HD before, during, and after sync. If you do it on every Mac then you'll keep your Desktop in sync between them all (files added, deleted, changed etc); this might not be what you're after though.
If symlinking isn't something you know how to do, instructions are below.
Backup (via TimeMachine or whatever you use).
Then fire up Terminal and run the following to move your Desktop folder into your iCloud Drive:
mv ~/Desktop /Users/<yourusername>/Library/Mobile\ Documents/com~apple~CloudDocs/
Then just symlink it to the usual 'Desktop' location:
ln -s /Users/<yourusername>/Library/Mobile\ Documents/com~apple~CloudDocs/Desktop /Users/<yourusername>/Desktop
Now anything that you add to ~/Desktop (by dropping things onto your desktop etc) will actually be dropped into your iCloud drive. They are of course still taking up local space on your machine. When you delete something from your desktop then it's gone from the 'Cloud' as well. The benefits of this are more for sync between machines (+ access via your iPhone of course).
If you've got another mac, move anything you value from it's desktop folder into your iCloud Drive 'Desktop' folder first, then delete ~/Desktop on that machine and symlink again as above.
Best Answer
Yes and no. The no is Apple is all or none with iCloud sync of files - you get documents and desktop sync when you enable iCloud sync on macOS.
But you can make a new folder called local Documents or whatever and move all the files you want locally to the new folder, then enable cloud sync. Your Documents proper folder is in the cloud as is the desktop, but the Documents will be empty other than what you intentionally want to sync up.