Thanks for your comprehensive report, wiz
There were some issues:
* The phase dialog for cards played as frogs showed "Scheme" cards, which were not in the game at all. (I assume that happens because the "Way of the Frog" is internally implement as this card, but in the interface it's confusing.)
You are quite correct, the Scheme picture was available and is used as a (confusing) shortcut to indicate Way of the Frog usage.
* When we played one of the Peasant etc. cards as Frogs, we could still upgrade them in the phase dialog! This makes for a quick way to get a teacher, but I don't think it's intended, is it?
This is correct and intended, and I don't see how this in any way makes you get a Teacher quicker than without Way of the Frog. Of the 2 effects only one can actually happen: Either you topdeck the Traveller for next turn, or exchange it for the next higher version. Because both involve a move of the discarded card from the discard pile to somewhere else, the stop tracking rule prevents the later effect from working, because it can't move the card anymore because it can't "find" it in the discard pile anymore.
Whatever of the 2 effects you put on top in the Cleanup dialog, will happen first and then invalidate the one below. Of course, if you select "Keep" for the Traveler exchange effect, the order shouldn't matter in any way.
* Later we got +cards, +money from the Teacher, but then we had to confirm playing each of these cards as "play them as card, not as frog" even when we clearly did not click on the frog part of the card (I guess the +card etc. is internally implemented as a "card" that can be frogged?)
This is due to a rarely relevant part of the game rules. The question is "When exactly do you have to decide if you play a card normally, or using a Way?" The answer is not
at the moment you play it, but after all the "first do something" effects that trigger on playing the card have happened. The most common ones are opponent's reaction cards to you playing an attack, and the +1 something tokens from Adventures you mention that Teacher gives out.
So for example, you play Sea Hag, indicating that you do not want to use a Way. If no player reacts, then you do the normal effect of Sea Hag, your opponent(s) get a Curse to the top of their deck. Now assume all your opponents reveal a Moat. As I said above, the game rules allow you to make the choice to use/not use a Way after the time your opponent reveal/don't reveal their Moat. So the online implementation asks you to reconsider. For Sea Hag it makes total sense to use the Way now, because a Sea Hag does nothing for you (and now nothing against your opponents).
At other times, it may be unclear: Do you want the +$2 from Mountebank, or would you rather take your chance with Way of the Pig to get another card, if your opponents all Moat the Mountebank? If 2 of your 4 opponent's Moat your Sea Hag, would maybe the +$2 from Way of the Sheep be better? So the game asks you to reconsider.
The same happens with the +1 something tokens from Adventures, because they also work "first" when you play a card from a pile that has the token. If you have such a token on the pile of a card, you get asked to reconsider your decisison, no matter what you initially selected.
Now, that seems rather pointless for the +1 Action, +1 coin and +1 Buy tokens, because there is no suprise factor here (different from you playing attack cards and you not knowing if your opponent have Moats). So there is in Autoplay that suppresses the reconsider question for those tokens. You can find it in the Autoplay options, under the "Default autoplays" section, as "Champion" (because Champion gives you +1 Action "first" every time you play an Action card). If you change the option to "Smart reconsider ways", the game will not ask you to reconsider for those 3 tokens. If you are in the game, right-click any card that gives out such a token (like Teacher or the Event Seaways) and select (for this game) "Smart reconsider ways".
But for the +1 Card effect, there is a surprise for you, because you don't know what you'll draw. So in this case the game always asks you to reconsider using/not using a Way.