Hi there and welcome
Can you please hold the home and power buttons on the iPhone for about 10 seconds until you see the apple logo then release.
Then let the iPhone restart and tell me if it then functions correctly please.
Can you go to settings, general, reset, reset all settings.
Once done, tell me if it then clears up the issue please.
Can you go to settings, general, reset, reset network settings.
Then tell me if it downloads please.
can you restart the computer please then try.
Can you connect it to a different USB connection on the computer and tell me if it then syncs please.
on the iPhone, go to Settings > General > Usage > Storage. Under Storage you will see how much space you have left on your device.
How much space is available?
In iTunes when you select your iPhone, go to the Music tab - is "Entire music library" selected?
set it to add all music please.
you can play the songs within iTunes correct?
ok, you will need to reset the phone, this will remove all data from it and return it to new so please backup any data from the phone first
Then go to settings, general, reset, erase all content and reset settings.
Once done, setup the phone again and tell me if it then functions please.
No problem, let me know please how it goes