I have played the whole game while live-streaming on Twitch.

This is basically already perfect. The minor gripe here is that the way you said it, makes it sound like you've played the whole game more than once. These
Alternate versions provide a tiny bit of extra clarity:
* I have played the whole game, all while live-streaming on Twitch.
* I played the whole game while live-streaming on Twitch.

The previous game “Tales from the Borderlands” i, was developed by Telltale.

The comma is optional

This new onelatest entry in the series is developed by Gearbox.

I felt that the original way you wrote it was a little too much on the conversational side of English.

I loved the previous game by Telltale.

I might be splitting hairs here but you might want to past tense up "love" on this one.

I had so much fun with the entire gameall throughout the previous Tales from the Borderlands, so I really love this new one too despite its mixed reviews on Steam.

This is all style change.

I had to change things around to fit everything into one sentence here. You do need to repeat the full names of these games a lot since the names are are so long.

TWhen playing through the game there are some serious moments, but almostotherwise the entire game is comedy.

So you're hopping into a new subject in this sentence, you'll need to preface this to the reader.

The parts I changed after the comma is almost exactly how a native speaker would sound if they were talking. The problem is that we're writing here so we need to be a bit more concise.

I love the humor in the game, so I laughed a lot throughout the gameplaying it.

just tidying things up so that it reads 100% like a native wrote it.

Almost allNearly all the scenes and conversations makde me laugh.

We're were talking about playing the game in simple past tense in the previous sentence. It is a little jarring to switch to present perfect in this sentence.

Check out the related link in the notes section.

Hey you nailed this one, I remember mistakes on the last time you wrote this part. I can see you've improved, that was fast.

The previous game, “Tales from the Borderlands,” iwas developed by Telltale.

I had so much fun with the entire game, so I really love this new one, too, despite its mixed reviews on Steam.

There are some serious moments, but almost the entire game isthe game is mostly comedy.

Almost all of the scenes and conversations make me laugh.

It’s a narrative oriented spin-off of Borderlands 3, and there is noeither action nor shooting.

or: there is no action or shooting

sounds weird to me to hear no & nor together. i always hear neither & nor, or no & or. perhaps that's regional?

I love the humor in the game, so I laughed a lot throughout the game.

you've used the word "game" 5 times in the past 4 games - i would say "i laughed a lot throughout it" at this point. or "i laughed a lot while playing"

I have played the whole game while live-streaming it on Twitch.

The previous game “Tales from the Borderlands” iwas developed by Telltale.

This new one iwas developed by Gearbox.

'was developed' (complete product) vs 'is being developed' (features still being added)

There are some serious moments, but almost the entire game is comedy.

Stylistic rephrase: There are some serious moments, but the game is almost entirely comedy.
Sounds more fluid to me, but the original isn't wrong.

I love the humor in the game, so I laughed a lot throughout the game.

Almost all scenes and conversations makde me laugh.