It sort of sounds like that you don’t need JupyterLab. It sounds like you want Jupyter kernels. I think Juniper may be what you want to tap into.
But maybe actually you just need nbconvert on your server and you can programmatically extract results from notebooks to your apps’s frontend. See here for how you can execute notebooks from the command line, selecting which kernels get used to run them.
Related to all this, have you looked at Netflix’s papermill and scapbook?
As for how the communication is done, I cannot help you there. I’m not up on what is under the hood or on the back-ends. I would think you’d want to explore more posts in this discourse to get a better sense. Also there are some community meetings that get posted about here. One of them may allow you to connect with someone that can point you in a direction to the documentation you seek. One thing to point out is that JupyterLab 3 is very new so you don’t want to look at older stuff.