Question - Live Streaming between 2 laptops using Cat5e cable

Greetings all.

I hope this is the correct place to ask this.

I'd like to run a game application which has keyboard inputs for up to 2 players. I was wondering if, instead of both users using the same laptop and keyboard to play the game, it would be possible/doable for one player (host) to live stream the game application from their end to the screen of the other player (client), connecting both via a Cat5e cable? The game application itself is not a heavy title like COD, BF etc., but rather just an application made in another software.

Additionally, is it possible to also have the client's keyboard sync with the host's, so that if the client presses a key, that key would register on the host's end (keeping in mind that the keyboards are fixed to the laptops and not detachable)?

I've tried googling this question but my search results only yield related subjects, rather than directly answering the question.

If anyone has had experience with this, I'd love to hear what you did to make it work.

Thanks!