Are all public-key encryption protocols based on one-way functions? Are there any public-key cryptography protocols which don't rely on one-way (or trapdoor) functions?

RSA and Diffie-Hellman cryptographic protocols both rely on one-way functions (prime factorization and discrete logarithm). Lattice-based cryptography is based on the Shortest Vector Problem which is NP-hard, but is it a one-way function?
Yes, all public-key cryptography based on one-way functions. Every one-way function builds on some hard problem. There is many one-way functions in lattice-based crypto, that based on SVP. For example, NTRU one-way function, LWE one-way function, etc.