if I understand well : private key is a combining of random number between 1 and 10^77.
From this private key is derived a bitcoin address with a precise algorithm.
As long as you send BTC to this bitcoin address which has been derived from your private key : you can safely say that you have the control of your fund and that your private key is valid.
can someone confirm?
Correct.
If I recall correctly you can actually generate two bitcoin addresses from a given random number. But it's certainly correct that the process is completely specified. There's a python library that you can use to play around with keys from numbers, maybe 'ku' (key utils), don't remember. Anyway, python is convenient for learning, I think, the code is pretty readable and you can use the REPL to explore interactively.
Yes, the example in the quote below shows the two different valid addresses derived from the same private key (extremely long random number).
The first notepad screenshot is of the extremely long number (in hex format) that the two Bitcoin addresses were derived from.

This is the private key copied from notepad and pasted into an offline copy of the bitaddress wedsite. Click the view details button to get the private key converted to normal formats.

This is my multibit wallet's address 1F84fkbMng6dJpGZmtycRbUe72B7XSYbeT shown on the right hand side of bitaddress. Every raw private key can convert into two different bitcoin addresses, which is why there's two.
