Is it possible create a fake transaction from a BTC address to another without paying any fee and also without this transaction gets ever confirmed?
There is nothing like a fake bitcoin transaction been sent, this is only possible in if the bitcoin network gets compromised and you know how hard it is to do that. The other reason is if you broadcast a fake transaction it will be dropped or maybe except you have your own node or something like that and it is added through the back door.
I will explain the point.
for exemple I have a BTC wallet with 0.00001 BTC and I just want to send 0.00001 to another wallet in order to this wallet just see the incoming transfer but I want it to get rejected by blockchain, like a fake node input on blockchain, is it possible?
The smallest unit of bitcoin that cannot be send is something below one satoshi which is 0.00000001btc. So even if you send a transaction like 0.00001 containing even the fee and the mempool gets way less congested theoretically, it can be spend. The only way to get it dropped is by double spending with a higher fee or the it so dropped after 14 days due to low transaction fee