MacOS closes all the applications when I lock the screen

I have not found a way to disable filevault auto-shutdown, but I have found a way to restart while bypassing the filevault pre-boot login screen (where the auto-shutdown happens). Entering this command in terminal will allow a remote restart without the risk of a shutdown:

sudo fdesetup authrestart

After entering the command and a password for sudo, you will need to enter either the filevault recovery key or a user password for a filevault-enabled user. This password will then be stored in memory and used automatically upon boot to satisfy the filevault authentication requirement. After the disk is decrypted, you will then be taken to the standard post-filevault login screen which doesn't trigger shutdown after 5 minutes.

Additionally, the authrestart command could be set to load a plist file automatically each time the computer is booted to authenticate filevault automatically, but this partially negates the purpose of filevault, and I don't think it's a great idea to store a password in a plist file.

I found this information here.

I'd still like to find a way to disable auto-shutdown permanently, but at least this is a workaround for the remote restart issues.

If the issue occurred after an update you may have a potential software issue. I understand you have tried a lot of troubleshooting in the comments but I listed the steps that I would follow below.

1. Make sure you backup all your files first before continuing troubleshooting: https://support.apple.com/en-us/HT201250

2. Follow this support article from Apple regarding video issues using external displays: https://support.apple.com/en-us/HT201177

3. As @Stefan suggested, Try reseting SMC. Shut the machine off. Press Shift-Control-Option on the left side of the built-in keyboard, then press the power button at the same time. Hold these keys and the power button for 10 seconds. Release keys. Power Machine on. SMC: https://support.apple.com/en-us/HT201295

4. Try reseting NVRAM: Shut down your Mac, then turn it on and immediately press and hold these four keys together: Option, Command, P, and R. You can release the keys after about 20 seconds, during which your Mac might appear to restart. On Mac computers that play a startup sound, you can release the keys after the second startup sound. NVRAM: https://support.apple.com/en-us/HT204063

5. Try booting in Safe Mode as @Allan suggested and see if you can reproduce the issue. Restart machine and hold down the shift key till you see login screen. Safe Mode: https://support.apple.com/en-us/HT201262

6. Run diagnostics on your device locally and check for potential hardware issues. Restart machine and hold the "D" key till prompted to run diagnostics. Even if you machine passes the diagnostics, you may still have a hardware issue. note the Apple Store Genius Bar can run deeper tests on your machine if you schedule an appointment. Diagnostics Mode: https://support.apple.com/en-us/HT202731

7. If hardware diagnostics passes, you can use Recovery Mode to reinstall the operating system or erase the machine and do a clean install. In recovery mode, if you choose to reinstall the macOS it will preserve the files on your system. You can also perform a clean install from macOS by reformatting the storage drive before installing the macOS. Recovery Mode: https://support.apple.com/en-us/HT204904

8. You can try updating to the latest macOS. I was having issues using display port and external monitors, updating to High Sierra resolved the issue for me. Update the software on Mac: https://support.apple.com/en-us/HT201541

At the end of the day, the issue you are having is either software or hardware related. By performing a clean install, we eliminate any potential software issues, not factoring in bad display driver code (typically these drivers get updated when you run system updates or upgrade macOS). If the issues continue to happen you may have a hardware issue of some sort either with your MacBook Pro, the displays, or the cables.