Debug console doesn't report problem.

[hongyi-zhao]

Try with the following code snippet:

<< JavaGraphics`
g[x_,y_]:= 6 Exp[x^3] Sin[y]
Plot[g,{x,-20,20,{y,-20,20},PlotRange=All}]

In vscode:

In MMA notebook:

[kenkangxgwe]

It doesn't because I haven't figure out how to capture the messages during evaluation in another kernel. But if you open the log panel, it will emit something.

[hongyi-zhao]

But if you open the log panel, it will emit something.

Yes. As shown below:

[kenkangxgwe]

I think this can be partially done by wrapping the evaluation with EvaluationData, but I cannot keep them in the same order as FrontEnd. However, it's still the most worthy approach that I can think of. Same for reporting messages.