Great, was able to sent some arte to etherdelta. So if you like to buy some arte, go ahead!

 1. do I need in some way to start the process of self-mining on the genesis address?
2. (how) can I transfer ARTE self-mined by a genesis address, when my address already shows the total amount bought in the ICO?

1. Nothing, it will mining by itself
2. go https://etherscan.io/token/0x5f6e7fb7fe92ea7822472bb0e8f1be60d6a4ea50#readContract copy paste your gen address at availableBalanceOf click query and divide the result by 1000000000000000000 thats your arte available for transfer

Did the source code for this ever get released?

What's the advantage to having a genesis address? Just checked and have mined 2.62 already. 

I checked now and i received Arte on my address. But how to calculate how much balance is available for send?

I see the total ICO ARTE in my MEW and Etherscan. I don't see the mined ARTE in the genesis address. Do I need to do anything? Will the mined ARTE automatically appear?

https://www.artemine.org/tutorials

There are tutorials about everything which is related to this token so simply click on Check available balance and follow the instructions. I hope you will see the balance how much available right now.

Thank you dev for these very hepfull howto.

For the public mining :
 - I have to send my ARTE to a non genesis ARTE address ?
 - Then, I can call the public mining function and get 0.00003200000000 ARTE (3200000000*10^-14) for each ARTE in my wallet ?
 - Then, after each call the return is 0.00000000000001 ARTE less (10^⁻14) ?
 - And I can call the public mining fuction as many times that I want (continuously with a script for example) ?

You'll need to add the custom token to myetherwallet or whatever you used:

Contract Address: 0x5f6e7fb7fe92ea7822472bb0e8f1be60d6a4ea50
Symbol: ARTE
Decimals: 18


Hope that helps!