You didn't say anything about your general level of experience with programming.  But if you don't have much general programming experience I would suggest sticking with a pre-built solution.  GMnet seems like the easiest of the above choices to see some return on.  And, yes it looks like it will only work with GameMaker.

I strongly recommend reading everything that relates to multi-player gaming on this site: https://gafferongames.com/

Even if it's not directly related to what you are doing, the explanations and diagrams they use are top notch.  They will give you a much better understanding of the task you have ahead of you, and may help you make a more informed decision about which route to take.

Best of luck!

There are several kinds of game coding, really.

There's engine coding, which builds "networking" and "graphics" and "sound" and "art tool plugins" and so forth, typically in C/C++.

There's "scripting" coding, which builds GUIs and game logic and so forth, typically in some scripting language like Python/Lua/Blueprint or something like it.

There's "shader" coding, which builds the specific 3D light/shadow/material solutions, typically in shader code like GLSL or HLSL or in shader graphs for specific engines.

Even within each of these kinds, people often end up specializing as budgets go up -- there are people whose only job is to make fire and smoke shaders for FPS games!

So, "which kind of coding" should you learn? It depends on what you want to do, and which game engine you're using.

By the way, the simplest possible networking model for a party game that's only played on a local network, is to have each player send their actions 20 times a second using a UDP broadcast packet to a known port. Make each game client listen on the same port, and assume that any packet received is from another player. This would make it trivial to cheat, but party games aren't particularly susceptible to this. And the cool part of this is that there's no match-making needed, no server needed, no set-up needed. Just start the game, and you end up joining in with everyone else!

No, for online play you need a client/server architecture.

Go with an existing game engine that has networking built-in would be the easiest.

In general, gameplay implementation needs to take networking into account from the ground up, it's not usually possible to "bolt on" networking afterwards, which is why using an engine with networking from the start is a good idea.