We know that the Bitcoin currency is volatile, but does the transaction fees correlate with it's volatility? Is it right to assert that if there is high level of Bitcoin demand, then the currency goes up, and then if the demand comes from a great amount of person, then the network size increases and so the transaction fees could be balanced by a greater availability of blockchain space, or am I going wrong ?
No, the more people use Bitcoin, the more the transaction fees will go up. This is caused by the fact that the blocks are generated with a 10 minutes average and they can only hold up to 1mb of data. The more people use Bitcoin, the more there's a demand to have their transaction in that small 1mb block. This is why there was a fork for Bitcoin Cash which advocates bigger blocks (Currently 8mb, but they will scale with the demand to keep fees low).