If your laptop is powerful enough to run vMix then you should be able to stream from it too.
but not knowing the total picture it's hard to advise you on the best way.

if your computernis powerfull enough it should work on one comuter. however, i sometimes split into two computers aswell to seperate mixing and recording and streaming. in this way i can also use different titles and banners etc.

i use a bm card or shuttle as an output and input this in the streaming computer. thus you can control everything and seperate.

using a lan cable is possible if you run the vmix supplied desktop capturing program and capture in the second screen. i would not recommend this! i sometimes use this option as a quick and low end way to capture a camera or webcam connected to a second computer or laptop and capture the output as a desktop capture. p.e. i do this to connect an extreme viewpoint, from the top of thevceiling or far away in the other end of a stage. the quality of the image and framerate is not so important.

for livestreaming just try your luck with one computer and reduce quality and resolution when you reach maximum capacity.

good luck...
stfan