Network Connection Timed Out

If you want to do a test to narrow down the cause, create a new "Test User" account in System Preferences Users & Groups pane. It can be Standard (not Administrator). Log out and log in to new user account (do not use Fast User Switching for this purpose). In new user account, run iTunes and sign in with your Apple ID. You have a new iTunes library there, but you should see your iCloud Music Library. You cannot sync your iOS device, because it's a different iTunes library, but try doing other things in iTunes that cause the observed problems.

If those problems do not occur in new user account, the cause is probably in your usual user account. For example, you may have an incompatible login item or iTunes plug-in, or a problem with your usual iTunes library. If those problem do occur in new user account, the cause is probably at the system or application level. For example, an incompatible system extension, or some type of iTunes "bug," or a network connection issue. Either way, you can better focus your troubleshooting.

There has to be at least ONE Administrator user account. That's probably the one you are logged into right now. And it does have a password, but you can have it set to do Automatic login. So you don't have to enter it when you start up your computer. And if you have the Security & Privacy pane set to not require a password when waking the Mac, you don't need it there either. However, you do need the password to do things like update software or make system (not user) level changes to settings.

Open System Preference Users & Groups pane. In the sidebar (left side), where it says Current User, THAT is your user account. That preference pane makes system level changes, so you need to click the Lock symbol and enter your administrator password to unlock the settings. If you're saying you don't know the password for your admin user account, that's a problem. Not just for this one purpose, but for other things like authorizing installation of third-party software and changing other system-level settings.

You won't be able to add a new user account, unless you can enter the admin user password on the Users & Groups pane to unlock it.

Yes, you can remove the "test" user account after you're done with it. Actually, even with that other account present, nothing changes regarding your current user account. It is still set for automatic login and the security setting for no password to wake is not affected. It sometimes helps to have the second user account, for testing and troubleshooting, so you may want to keep it around. It does not take up very much storage space, since you're not actively using it. But it's like having a second Mac that you can switch to, with its own user-level settings and data... (You should make it Standard, not Admin.)