Is really a random number of the ECDSA secp256k1 so it has an specific protocol and is not exactly a random number?
No specific protocol. It is just a random number. Some people "choose" a number by using some memorable data and running the SHA-256 algorithm against that data to result in a number ot use as a private key. This is often referred to as a "brain wallet". By using this "brain wallet" technique you drastically increase the odds that someone else will pick the same data (or guess what data you've picked), so this is generally considered insecure. The most secure way to choose a private key is to do it as randomly as possible. There are cryptographically sound random number generators used by computer programs (wallets) to do this, but if you really wanted to, you could just flip a coin 256 times and write a 0 for each "heads" and a 1 for each "tails".
There was a sinple equation or way to calculate the whole public key form the private key, or if it's complicated explain in a simple words how this conversion works? The private key is hashed with something or is a simply aplication of an equation?
As stated in quote you found in a forum, it is "an Elliptic Curve point multiplication with the curve’s base point. The result is an (x,y) coordinate pair, which constitutes the public key."
No hashing, no other equation. You'll want to learn about Elliptic Curve mathematics, and then you can perform the math yourself (although it would be far safer, faster, and less error prone to use a well tested and industry accepted library instead).
