OK, need a little more information:
1. You indicate that you have a new phone that you need to activate. Then you say you can't activate it until you back up photos to iCloud. Can you explain the connection between those two sentences? Do you mean you want to restore the new phone to a backup from an older phone, but that the backup for that older phone is too large?
2. When you state that you want to back up photos to iCloud, do you mean you are using iCloud Photo Library? Because if that is what you mean, then you don't back them up, they sync automatically. However, if you mean that you want to store photos on that other phone somewhere so you can delete them from that older phone, then you don't back them up to iCloud, you Import them to a computer: PHOTO IMPORT IOS TO MAC/PC. Photos that are on iCloud are in the iCloud Photo Library, and if you delete them from your device, they get deleted everywhere.
3. The third issue does not seem related to the other two at all. You are indicating that you have an older version of your Apple/iCloud ID still signed in to iCloud. If that is the case then you need to do one of two things to get that older ID signed off and the newer one signed in:
- If the Apple account for the old ID is the same Apple account that is being used for your newer ID (i.e., if you simply went to Apple - My Apple ID and changed the ID you were using to the new one, then all you have to do to sign the older version out is to go to Settings>iCloud, scroll to the bottom, and click on Sign Out. When it asks for your password, type in the current password you are using for that ID.
- If you set up a brand new Apple account for the new ID, then you will have to go to Apple - My Apple ID, sign in with the old ID, and change the password so that you can then sign that old ID out. If you don't remember the answers to your Security Questions, you will have to contact Apple Support to have them send you a reset link: ACCOUNT SECURITY SUPPORT. Be aware, that if you did, indeed, create a totally separate ID, then the iCloud information saved under your old ID will not transfer to your new ID, so it will become critical to import your photos. You would also want to select to Keep your Contacts on that phone when you sign out of the old, separate iCloud account so that you can merge them with your current one when you sign in.
Best of luck,
GB