I find DL is happy with both forms when translating from spanish. When translating to spanish it is a bit inconsistent. Quite often it will accept vamos a encontrar, but you are safer with encontraremos if you are asked to translate "we will find". I think vamos a encontrar is called phrasal future (we are going to find). Encontraremos is (we will find) future, not sure what its proper tense name is.
If both statements are of determination, but not of certainty, they could be considered equivalent. Yes, I would probably comment if my translation as "we will" was rejected, but I do see your point of view on this. Again, this would depend on the context in which it is said and what degree of humor or bravado is in the speaker.
I don't think it would be a big deal adding "these" as well as "those" keys... Just got a wrong answer because of that, even though the meaning of "esas" can mean "those" and "these"... Just saying... I know 'We're gonna find "those" keys' sounds better, but sometimes it's just automatic and instead of those you put these, it's a common typo even... You should take Norwegian as an example, lots of typos and similar translations added :D
Right context maybe, but it's at least a wrong grammar. "Let's do this" is a rather fancy form of the imperative mood. It's more a demand/suggestion than the indicative "This is going to happen".
I would translate your sentence into Spanish as "Vayamos a encontrar esas llaves."