First Question: private key for what exactly?
For the scriptPubKey that locks/encumbers the coins from being able to be spent.
An output contains instructions for sending bitcoins. Value is the number of Satoshi (1 BTC = 100,000,000 Satoshi) that this output will be worth when claimed. ScriptPubKey is the second half of a script (discussed later). There can be more than one output, and they share the combined value of the inputs. Because each output from one transaction can only ever be referenced once by an input of a subsequent transaction, the entire combined input value needs to be sent in an output if you don't want to lose it. If the input is worth 50 BTC but you only want to send 25 BTC, Bitcoin will create two outputs worth 25 BTC: one to the destination, and one back to you (known as "change", though you send it to yourself). Any input bitcoins not redeemed in an output is considered a transaction fee; whoever generates the block can claim it by inserting it into the coinbase transaction of that block.
which system validates that I am a private key owner?
When transactions are submitted, nodes on the network validate the signature with the scriptPubKey before including them in their mempool for miners to include in the next block.
Now these two amounts of bitcoins are taken by a system X (what is X?) and it will be created an output with amounts 2 Bitcoin (i) and 0.8 Bitcoin (j).
The transaction is created and signed using wallet software, and the transaction is submitted to the nodes on the network, it is propagated to all nodes (given it passes the validation checks).
Where will the public and private keys for i and j are generated? How and from which holder (after generation) to which (new owner) will they be transmitted? What kind of message do they sign/encrypt (content).
The private keys are generated by the user usually in their wallet software and usually the pubkey and scriptPubKey (which is encoded into an address) are also created by the user for people to send them Bitcoin. The message that is signed is the new transaction data itself.