The idea is good.
What you seem to be doing is looking for a way all through to the end. Maybe very very few people have done it. So it would be hard to see all the way.
The project that gets started is the one that progresses.
So, I encourage you to get started and believe in yourself.
At least you will reach somewhere if not all the way...
Maybe you will come up with your own solutions to small problems that hamper progress...
Now, as someone who has built an htpc on windows, I can offer some things...
With so many functions that you have in mind (remote, calibration etc), right now instead of diving deep into just one area, you might want to focus on playing the role if a system integrator...one who gathers individual solutions made by others and integrates them to work in a system. By this itself you will learn a lot. And this itself will take some time and its already quite a goal.
And then there will be the cost thing...
So why not first build a basic htpc and then in second version go for the full thing...
What an avr does is 3 things minimum
1) Decode formats - DTS, Dolby etc
2) DAC - of the streams in there - 6(5.1) or 8(7.1) or more
3) Amplification of those streams - so a separate amp for each stream.
To manage within budget, a media centre will do good...
It will do the media mgmt + playback + decode functions well. Additionally the headache of getting latest linux based codecs of dolby, dts etc plus integrating them into playback software will be handled by the media center.
Yes there are free codecs that have been reverse engineered for dolby, dts etc even for linux platform.
(I dont think there will be too much difference between software decode and hardware decode in terms of quality...data is not lost in a proper software decode esp if its a licenced one from dolby etc...what is missing is a good 7.1 channel dac)
So, the solution is known more or less...
1) Media Center for playback...I used Nero as it is cheap plus picture quality was good plus it uses licence from dolby etc. It didnt have support for 7.1 3 yrs back though. (Check its MediaHome software). Additionally it offers streaming music from android phone to pc via wifi ... that was good.
For linux check link...
https://www.maketecheasier.com/media-center-linux/
2) ASUS Xonar DX for dac part...it has good quality and affordable 7.1 decoding. Additionally it has dolby conversion options...7.1 to 5.1 to 2.0 and reverse. Check ASUS Xonar D2X for similar DTS functions. But even nero can do them internally I suppose.
3) Amplification was combined with spkrs and I bought affordable 5.1 spkr set with built in amplification.
The problem i had was that remote control integration was hard and i was not satisfied with the solutions available then.
For individual pc components selection and case (very imp for the look...) check my detailed post here some time back....
http://www.hifivision.com/home-thea...s-all-i-have-learnt-about-building-htpcs.html
Take your time to go through the complete guide...its quite detailed...and I had worked a lot on it....more than most guides on the internet....and it gives fundas...so its relevant even today
I hope you find this useful
Best wishes
(Addendum...720p h.265 uses 30% cpu on potplayer on my i3-3220 system. Audio decoding doesnt use much cpu resources. Video codecs come around once in a decade. So if you go for todays gen i5 system then it should easily last a decade...even for 4K h.265)