Implication of solving 3SUM problem of a certain size on the Exponential Time Hypothesis
I think currently it is not even known if strong ETH and 3SUM are related, see e.g. [1]. For the relation of ETH and 3SUM, note that ETH really cannot be refuted by improving polynomial time algorithms (at least via Karp reductions) because it would only improve constants in the exponent of the runtime. In particular, if we reduce 3-SAT to a 3SUM instance of size $2^{O(n)}$, it will not refute ETH, and if we reduce 3-SAT to a 3SUM instance of size $2^{o(n)}$, it will refute ETH regardless of the complexity of 3SUM.
[1] Virginia V. Williams. Hardness of easy problems: Basing hardness on popular conjectures such as the strong exponential time hypothesis (invited talk). IPEC 2015. https://drops.dagstuhl.de/opus/volltexte/2015/5568/pdf/5.pdf