Actually it has three, ליגוע (but OK, that's a close variant of לנגוע, and I'm not sure it's considered standard; FWIW writing it seems strange, but I think it's often said).
I think it's just a quirk. לנגוע or ליגוע follow the normal pattern for this root (given it's initial נ), while לגעת follows a pattern of only a few verbs, generally not with an initial נ, but very common verbs. Apparently some speakers in ancient times drew a false analogy and it stuck. I can't see a reason why it happened here and not in other initial נ verbs. Maybe professional linguists do have a richer story, though.
It's not a literal translation - see my posts above and below. You're assuming that because ב is translated "in" when talking about position, it always means that and should always be translated the same way, but languages don't work like that.
Incidentally, even if we are talking about a verb that doesn't take ב as it's preposition, אתה would still be wrong, because that's the form of the pronoun you use when "you" is the subject of the sentence. If it's the object of the sentence and the verb doesn't require a preposition (just takes a regular definite object), it will be אותך.
You're making the mistake of assuming there's a clear cut one-to-one relationship between Hebrew and English prepositions; there is not. In English, we put something in a bowl, we touch something (no preposition) and we watch TV (no preposition). In Hebrew, all of those actions require the preposition ב;
יש משהו בקערה
הסוס נוגע בך
אני צופה בטלוויזה
It's just the way the language works. There's no particular rule that a given preposition should match exactly between two languages.
To take an example or to from English, in British English, they write to someone, but in American English, they simply write someone. To British ears, writing someone sounds very weird - we write a letter or a book, not a person. But to American ears, the preposition isn't necessary. If we branch away from prepositions, take the word "tap" - for Americans, as far as I'm aware, the noun version would only mean what you call it if you briefly but sharply touch something: I gave him a tap on the shoulder. But in British English, it also means what Americans call a faucet.
And that's just two varieties of the same language!
With two languages, especially if they're not related, there's no guarantee whatsoever that a word that has a given array of meanings in English will have one equivalent with the exact same set of meanings in Hebrew. In fact, it's far more likely that they won't. This, incidentally, is one of the reasons automatic translation tends to be unreliable.
A reasonably good Hebrew/English example: In English, one plays sports and musical instruments, and acts in a play. In contrast, in Hebrew, to play sports and to act in a play use the same verb, לשחק, but to play an instrument is לנגן. The divisions between what verb applies to which situations isn't arbitrary, but it's not set in stone or the same between languages, either.
(For the most simple example, actually, look at the copula. In Hebrew, we say אני שרקן, אני אישה, אני מאנגליה. That doesn't mean that in standard English it's okay to say "I actor, I woman, I from England.)
TL;DR: The sooner you give up on the idea that any language is simply a set of words that corresponds exactly with the set of words and meanings in your own language, the less frustrating you will find language learning. Hebrew uses ב for watching and touching because that's how Hebrew works, and we translate it into English as a simple direct object because that's how English works.