Remove Photos Without Removing From iCloud. Upload not working

About iCloud Photos: Using iCloud Photos synchronizes your Photos Library with your iCloud Photos Library. Having iCloud Photos turned on makes the pictures in your phone's Library exactly the same as pictures in your Library at iCloud.com. As long as your phone is connected to iCloud, you can't remove pictures from your phone without them being removed from iCloud.

To save storage space you can turn on "Optimize Storage" on your phone. Optimize Storage saves space on your phone by keeping only smaller pictures there, and relying on the full files being kept at iCloud Photos. Then, when you need the full pictures for printing or editing, etc, the phone grabs the full file from iCloud. This is very cool, but it can lead to a slight delay, sometimes. With Optimize Storage, you save maybe 80% of the storage that would be used otherwise.

Do you have "Optimize Storage" turned on?

If you have run out of room on your phone, you may have to delete some things, maybe apps you can reload later, so there is room to move around in there.

For your last question: if you disconnect from iCloud, delete pictures from your phone, and reconnect to iCloud, then Photos will reload those missing pictures to your phone in order to make "the pictures in your phone's Library exactly the same as pictures in your Library at iCloud.com."