I used to code Monogame in VS Code. Both on linux and windows.
I would create the project with VS Community and then code from there in VS Code. Everytime I added a .cs file I would manually edit the .csproj so that it would be compiled.
These are settings files (launch.json and tasks.json) for VS code on Windows: https://1drv.ms/f/s!AuKSxKmHzxT3luNXknAi0LouKtKRIw (OneDrive.com)
I’m pretty sure I followed the steps from ZeroSigma here: Visual Studio Code and Monogame?