How is the x coordinate of a "point at infinity" encoded in a Secp256k1 signature?
Standard encoding of the point at infinity is a single byte of value 0x00 (it is defined as such as least in P1363, possibly also in X9.62). Other representations may exist (such as a lot of bytes of value 0x00), but, in truth, the "point at infinity" does not have well-defined X and Y coordinates.

In the case of ECDSA, you generate a random value k which must lie between 1 and r-1, where r is the (prime) order of the conventional generator point G. Therefore, kG cannot be the point at infinity -- if it is, then the implementation got it wrong, or the key is not correct. Since kG is not the point at infinity, then it always has a well-defined X coordinate, and the problem does not arise. Similarly, if the point at infinity is obtained during the verification algorithm, then something definitely fishy is happening, and therefore the signature should be rejected (this cannot happen with a valid signature).

The same can be said about elliptic curve Diffie-Hellman. To sum up, if the encoding of the point-at-infinity begins to matter, then something went wrong and the best thing to do is to declare a failure (e.g. reject the signature as invalid).