My iPhone 13 won't turn on

The "Up, Down, Side" instructions require some precision:

• Press and quickly release VOLUME UP

     then

• Press and quickly release VOLUME DOWN

     then

• Hold the SIDE button until an Apple logo appears (15-30 seconds), ignoring "slide to power off" if it appears

The operative word here is QUICKLY; and try it a couple of times, the timing is a little tricky. If done correctly this will always work unless the phone has a hardware problem. 

1. There are many possible reasons it happened, from a failing battery, an iOS crash, some other hardware problem, water damage…From the other symptoms you mentioned, most likely your battery is failing
2. The force restart is your only hope if you don’t have a backup
3. If the force restart works back up your iPhone to your computer, or spend the 99¢ for 50 GB of iCloud storage and back it up to iCloud, or sync the important data to iCloud:

iCloud+ has 2 independent functions. iCloud backups, and iCloud sync. It’s confusing because they both have the same name prefix. But they have no connection with each other. Go to Settings/[your name]/iCloud. You will see a bunch of switches (with iOS 16 or later also tap Show All). When you turn on a switch that data type will sync to iCloud. If you do this on multiple devices that share an Apple ID the selected data will sync to all of those devices. 

You can also turn on iCloud Backups. This will back up your phone every night if the phone is plugged in, connected to Wi-Fi and locked.

But note that anything that you sync by the first method will be excluded from the iCloud backups.

Nothing else you can do. Backups must be done before disaster strikes, after is too late. And there are no data recovery service that can get data off a disabled iPhone, as the FBI has discovered.