- What is the way to get the public key from the corresponding private key in Schnorr?
Multiply the private key with the generator G (like for ECDSA) and then throw away the Y coordinate.
Given the depth your questions go, I would recommend you to just read the relevant BIPs. In particular, BIP340 has a section on public key generation.
- What does that obtained value represent? Also point of the elliptic curve? If so, I assume its X coo, so how do we know corresponding Y coo?
Yes, a point on the elliptic curve, of which you only know the X coordinate. Internally in the BIP340 scheme it is treated as the point with given X coordinate and even Y coordinate, but at a more high-level you can also think of the public key as representing "either" of the points.
Given the curve equation y2 = x3 + 7, the corresponding Y coordinates for a given X coordinate can be computed as Y = ±√(X3 + 7) mod p. Note that this is a modular square root modulo the field size p = 2256-232-977. For the specific value p for secp256k1, this is equivalent to ±(x3 + 7)(p+1)/4.
This is not specific to BIP340 public keys. Even for ECDSA, typically public keys are represented in the so-called compressed format, consisting of sign_byte + [32-byte big endian X coordinate], where the sign_byte is 0x02 if the Y coordinate is even, and 0x03 if the Y coordinate is odd. Reconstructing the full point also requires a modular square root for this format.
- Does Schnorr (BIP340) also support a classic key tweaking like ECDSA, where we add some value to the public key and then add the same value to the private key and we would produce a valid signature?
Yes, Taproot even critically relies on this tweaking operation for representing script paths: in order to spend a Taproot output with point Q as a script, an internal public key P needs to be revealed together with a script (Merkle root) m such that Q = P + H(P || m)G. Since the Y coordinate of Q is not part of the output, it also needs to be revealed at spending time.
Does this mean that the number of possible points, i.e. the private key-public key pairs, is 2 times smaller than in the case of ECDSA since we only use even Ys?
Yes, but note that this is not a reduction in security (not even a negligible one).