To find the kashrus status of an organism, we first need to correctly classify it as plant, animal, bird, or asher bamayim*, as each group has its own distinct requirements. E.g., animals need to chew cud and have cloven hooves, while asher bamayim need (fins and) scales.
What is the halachic definition of asher bamayim (אשר במים)?
* This is how the Torah calls water-living creatures. Rabbinical writings use the word dag (דָג).
A preliminary answer: according to the Aruch Hashulchan (Y.D. 83:10) the sea-creatures (i.e. fish) that are permitted by the Torah are only those that are fish-like, in that they have bones and proliferate. Additionally, he writes that sea-creatures which look like land animals - even if they have fins and scales - are not included in the kosher כל אשר יש לו סנפיר וקשקשת במים.
Simply speaking, the Torah really means to include everything living in the sea. This is implied by the Ramban on Vaykira 11:10 who includes sea creatures with legs like those of land animals, and the Vilna Gaon (Keilim 10:1) that says that the Torah gives all sea-creatures have the same halakhic status, seemingly in all matters.
If צפרדעים are frogs, then my inclination would be to say that they too are sea-creatures despite being amphibious (based on Shemos 7:28) but I have no source
