Keeping a 50/50 mix of old and new sounds like a good way of balancing things - maintaining the increased complexity/subversion of expectations ("frig" and "froge" are wonderful ways of expanding the possibility space in ways that align with what's already been learned, but still represent a curveball along the way) where possible feels important too!
I have a few ideas, but I want to first acknowledge that my sensibilities and tastes (not to mention workload!) aren't guaranteed to align with yours, and I'd recommend trusting your own instincts since that's what brings the most intrinsically "you" qualities to your work! I try to share experiences rather than unsolicited suggestions, but since you asked, here are the notes and thoughts I made while playing.
So far as a "sticker book" type thing would go, I think I'd look toward something that presented as detail-dense scenes that the player could pan around and zoom in on to find places to put their frogs. My mind goes to Richard Scarry's busy little town illustrations and Where's Wally books (possibly Waldo for you?) and the way that noticing little details and character's activities brings a sense of life and vibrancy, though those examples are much, much higher density than would be a good fit for Frog Finder, I think.
I like the idea of being able to place a frog anywhere and that be fine, but have some positions that are a match for particular frogs - the snorkelling frog might fit nicely at the beach, for example. A visual effect, or some special music, or a different two frame animation when placing a frog in one of those positions could give an extra layer of exploration and discovery.
It'd make sense to have a place (a separate scene maybe?) for sharks too - players are going to accumulate them, and having somewhere to place them and reflect on progress feels like it could help reinforce the sense of having passed through hurdles and put a fun spin on what might be less-positive for some players.
You've already seen it, but I could imagine a moment like this hiding somewhere in Shark Town :D
For tracking times and scores, I'm a big fan of inviting players to reflect rather than compete. High scores and best times have limited value - once you hit your skill ceiling, you'll get lucky one day and that's the best you'll ever do. Leaderboards give you abstract numbers to compare yourself to that represent people you typically have no social context for and will never have an opportunity share experiences with - only a handful of players get to be in the top 10, and the personal value of a ranking diminishes rapidly for everybody below that.
In some of my games, I've graphed the past N playthroughs/runs to let players see how they've improved over time and think about what they could improve. I still show historic best times/scores, but give more emphasis to how current performance compares to recent performance, which I think provides more opportunities for players to feel positive about and motivated by their experiences.
I feel like there's room for a lot of additional UI responsiveness - hover sounds, maybe visual pops, twists, or scaling when mousing over tiles could help make everything feel a little more dynamic and interactive. This one in particular, I had reservations about mentioning. I feel like Frog Finder has a very deliberate level of fidelity/style of polish to it, and maybe those kind of additions could push against that identity a little.
I'd love to see the "progress summary" screens be interactive, either by allowing players to click to select a "slot" on the current row, or by clicking a continue button. The opportunity to "stop" and rest/think/step away from the mouse for a bit feels like it could be of value to some, and the ability to click through quickly might remove a friction point for players who get impatient.
I'd also love to see a "return to menu" option to allow in-progress games to be abandoned. I can't imagine that I'd ever want to not see a game through, but I could see players who're trying for "perfect frog" appreciate being able to restart quickly. If this also allowed an in-progress game to be saved, that could be useful for players who need to break up play sessions on longer/harder difficulties too.
Last but not least, if the sharks had a small chance to trip while chasing the cursor around and then quickly get back up to rejoin the chase, that would be super adorable.
Edit: Oh! At some point, I was wondering if a splash screen for a fictional frog-focused educational commission or frog preservation society might be a good fit for helping to give some additional tone/framing going in.