Whelp, I'll do my best to answer your questions, but this topic has been discussed to death on these forums. You say you've reviewed those discussions. If reading all those threads didn't answer your questions, it is unlikely there is anything anyone can add to it to give a more comfortable response.

...understand that both CC-BY-SA and GPL licenses are problematic to use in a close-sourced commercial game, because it is unclear whether the sharealike clause applies only to the art piece being used, or to the whole project, even though there is a legal separation between the source code and the art.

I don't believe that is the case for CC-BY-SA. I don't know for GPL. In consulting with my own attorney, the only way a project's code (closed source or otherwise) would have to inherit the CC-BY-SA license from art assets is if the code itself were a derivative of the artwork. I'm not even sure how you would take art and somehow transform and adapt into being functional code, but I suspect drugs would have to be involved. That being said, my attorney was discussing my own project and its very specific circumstances. Paranoia may compel you to seek your own legal counsel.

1. What kind of proof do I need that I received permission from the artist (in case there is a protest afterwards) ?...

None. You already have it. They released thier assets under the CC-BY-SA license. Beyond that, a simple instant message or email saying "sure you can use it in your closed-source project" is sufficient. Beyond that? If you don't trust the author to not go back on their word, and are not willing to accept the legal terms of the irrevokable license by itself, I'm not sure you're ever going to have a sufficient "proof". People can sue you for any reason. It doesn't mean they'll win, but a scenario where an author goes back on their word suggests they are out to get you anyway. No amount of agreement or proof will disuade such people from giving you greif. To my knowledge, that has never happened with authors on OGA. If it did, they would be banned and their artwork removed; We don't suffer copyright trolls here.

That being said, there are some authors who prefer that their CC-BY-SA assets are not used in commercial projects. The courts' interpretation of the license takes precedence, so any disagreement would be unlikely to come out against you, but as you may have noticed, we're more about respect for our artists than we are about "don't worry no one will sue you", so it's definitely a good idea to ask them outright. That isn't always feasible and it isn't required for the use of the assets, but if you're paranoid about that sort of thing, it can help assure everyone has a mutual understanding.

...I found in this forum that I need to sign a Release Agreement,...

Where? Where on these forums was that said? I think I need some context in order to answer that question.

... I would like to ask OGA for authorization. Does OGA have the authority to allow me to use the CC-BY-SA assets from the website for my game ? Or do I have to contact every artist individually ?

No, and No. OGA doesn't have that authority because OGA is not the copyright holder. However, you do not need to contact every artist individually because they have already given you permission to use CC-BY-SA assets from the website for your game. If you mean "allow me to use the CC-BY-SA assets, but under different terms" like CC0 or something, then yes you will need to ask each artist for permission to use the assets under any terms other than the licenses already listed.

Note that in the case of LPC assets specifically, most of the artists have already included OGA-BY as one of the available licenses. Any LPC assets that lists OGA-BY are free of this kind of confusion about Share Alike clause infecting code.

3. I have read that CC-BY-SA is safe for open-source non commercial games, but NOT safe for commercial games. Is it only because commercial games are close-sourced so they cannot release their entire project under an open source license ?

Again, where? Where did you read that? In all likelihood that is due to a misunderstanding of Share Alike affecting code when the code is super unlikely to be a derivative of artwork. Non-commercial games are just as capable of violating the terms of CC-BY-SA as commercial ones:

Firstly, it is possible to have an open source game that is still commercial. Minecraft java-edition, for instance is open-source. It is not openly licensed, but the source is easy to view. Yes, I know that isn't what people generally mean when they say "open source" but the fact remains open-source does not mean non-commercial.

Secondly, being open-source or even open-licensed does not make the project automatically compliant with the CC-BY-SA terms. If your game is licensed MIT, but you never indicate the art assets are CC-BY-SA, or if you release derivatives of the assets as MIT or BSD instead of CC-BY-SA, or if you fail to give proper attribution for the assets? Well, then your open-source-libre-license-non-commercial project just violated the license terms. 

Conversely, closed source commercial games are fully capable of being compliant with the license. If the source is closed but the assets (and derivatives of those assets) are openly available, clearly marked CC-BY-SA, and properly attributed, then all is well.

Hope that helps. If not, further questions are welcome.