There is a LOT of work that goes into what you are talking about. It is a stretch towards true whole house automation, and you will have to not only learn a ton about how to do this, but will need to be adept at programming proprietary control systems with enough logic to handle different types of events. You will also need to consider the cost vs. actual usability of such a setup that can do this.
Understand that HDMI doesn't push out surround sound AND stereo, you get one, or the other, so you will need to have a DSP involved which can downmix from HDMI surround sound to stereo for rooms which don't have surround sound.
You will also need to think about whether you really want the audio and video following you around your home as you move about using NFC (near field communication) which foot perfect accuracy to know you are in one room vs. another and how quickly you want TVs and audio to turn on and off as you move about your home.
This is, at the very least, a MAJOR task in home automation. Beyond what I've ever had someone do.
In my mind, this is entirely inconvenient, as I could get up to go pee, and it might turn off the system which a bunch of my buddies were watching. So, you'd have to program in overrides, which means that you have to TELL IT when you want it to not follow you around, then tell it when you DO want it to follow you around.
Is this all really easier than just pulling out a tablet or phone and just turning the audio on in rooms which you will be going into?
Also, if you are in the kitchen cooking or cleaning while the game is on, you may have a need to set the audio a good deal louder, relatively, in that location vs. the room you came from, so volume should be unique to every room, not tied together so that you don't get the individual level of control you want.
Then you come back to questions: What happens when you have guests and they want to watch something different in a room than what you are currently in? How do you manage them and their movements vs. you and your movements?
In my experience, a simple hard button remote control and a distributed video system from a high end manufacturer is the only reliable way to get there. Remotes for video rooms, and keypads on the walls for audio rooms. Just press a button as you enter the room and you get the audio from the source you want. Adjust volume right there to a level that's appropriate. Tie it in to Alexa if you want to raise/lower the volume a bit without touching a remote. When you leave the room, press a button to turn it off.
You can do all of this with motion sensors tied into NFC devices, but it's tricky. There's a reason why major control system manufacturers haven't pushed this out as an 'amazing new product' even though the technology has been capable of it for years. It's because it really is impractical. Possible, but costly and impractical in real world use.
If you live alone, with no friends, family, or kids who are in the home, then it may be somewhat feasible.
Oh, and if you are watching something that needs good surround sound, then you wouldn't want it in stereo in other rooms that don't have TVs in them at all.
A very real world option?
Headphones. Yes, that may sound like a weird option, but wireless headphones can give you good stereo audio wherever you are standing and you can adjust the volume as you like for your specific scenario. If you are doing dishes and your wife is vacuuming, you can still listen to the game with excellent clarity without blasting out the speakers in your home.
If you do want to go down this path, you will need to dig deep into systems which can downmix surround sound to stereo while retaining the original surround track for distribution to multiple surround sound rooms.
I accomplish this with a Crestron DM switcher. The one I have is a 16x16 matrix that can accept up to 16 different sources and feed them to up to 16 different rooms. Latency is just a few milliseconds. It is all based on a backbone of HDBase-T technology, which is pretty much industry standard. My system currently supports up to 1080p, with some options for 4K/30 sources. But, I could upgrade it with cards to allow for 4K/60 18Gb/s sources if I want to. All the inputs are downmixed to stereo with a local output that feeds into 3 16x8 preamps. So, that means I can feed stereo audio from up to 16 sources to 24 rooms.
So, while I couldn't actually imagine wanting my home to follow me around turning TVs on and off all the time, I can imagine making it possible for such a thing to happen if I wanted it to. Motion detection in each room along with NFC in each room. Sounds painful to program.
I wouldn't expect a ton of deep help on these forums into a solution because of how complex this specific request is. It is the type of work which might take a few hundred hours of programming to accomplish, and you want to do your homework to ensure you are getting gear that won't then require maintenance every other day just to keep it up and running.
PCs are not designed for home automation really, so I'm not sure what you would use specifically, but there are automation controllers out there that allow for DiY programming and maintenance. I'm just not that familiar with the models that allow for the deep level programming you would need to get into to make it all happen and work exactly right for you.