How to choose secp256k1 private key? The private key is any random 256 bit (but smaller than prime p) number or must be prime or other condition?
For selected $x$ can be found $y$ - decompressing key:
#! /usr/bin/env python3
import binascii
import math
p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
def decompress_pubkey(pk):
    x = int.from_bytes(pk[1:33], byteorder='big')
    y_sq = (pow(x, 3, p) + 7) % p
    y = pow(y_sq, (p + 1) // 4, p)
    if y % 2 != pk[0] % 2:
        y = p - y
    y = y.to_bytes(32, byteorder='big')
    return b'\x04' + pk[1:33] + y

print(binascii.hexlify(decompress_pubkey(binascii.unhexlify('0229b3e0919adc41a316aad4f41444d9bf3a9b639550f2aa735676ffff25ba3898'))).decode())
print(binascii.hexlify(decompress_pubkey(binascii.unhexlify('02f15446771c5c585dd25d8d62df5195b77799aa8eac2f2196c54b73ca05f72f27'))).decode())
print(binascii.hexlify(decompress_pubkey(binascii.unhexlify('0279be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798'))).decode())

Only last x,y pair fulfill (x3+7-y2) % p is zero. Two previous pairs are wrong?
You are confusing the concepts; the private key vs the public key. Your private key $k$ is an integer which is selected uniform randomly between $1$ to order of the base point $G$

Basepoint $G$ in compressed for
  0279BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798


The order $n$ of $G$, i.e. $[n]G = \mathcal{O}$, where $\mathcal{O}$ is point at infinity, or the identity element in additive elliptic curve group.
FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C 


Your (or any) public key is $[k]G$, which is scalar multiplication and defined as.
$$[k]G = \overbrace{G+\cdots+G}^{{k\hbox{ - }times}}$$
With add double-and-add it can be calculated very efficiently.
Your failures are due to your python coding ( that is off-topic here) where you want to recover the $y$ value from a compressed public key, not the private key. Rasmus Faber's has an answer in StackOverflow. That is the way you need to find it wiht Python.
Note: All parameters of the curve can be found on Certicom's SEC 2: Recommended Elliptic Curve Domain Parameters