How do you derive the public key from you random 256-bits integer? Once I've flipped a fair coin 256 times and have the random 256 bits integer that I'm going to use as my private key, how is the public key derived?

I've read:

Given a private key, how do you get a public address

And:

https://en.bitcoin.it/wiki/Address

But I don't find the answer to my question. I understand ECDSA curves are used but I don't understand which part is the public key and what's the relation between the (unique?) public key and the addresses which you give out.

If you convert the private key to a public key by performing a multiplication with the curve's base point, then it begs the question: what is the curve's base point?

Is the curve's base point part of the private key?

Which would mean that what is called the "private key" isn't really just a "private key" but more of a keypair from which you can, at any time, derive the public key?
The curve's base point is a well-known constant. Since Bitcoin uses SECp256k1, the base point is:

04 (uncompressed point)

79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9
59F2815B 16F81798 (first coordinate)

483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448
A6855419 9C47D08F FB10D4B8 (second coordinate)

(See page 15 of my link.)

The private key is an integer from which you can derive the public key and produce signatures that can be verified with the public key.