Probably no. I would translate "I can't see you" as «Я тебя́ не ви́жу» normally.
«Я не могу́ тебя́ уви́деть» is a possible alternative, but it implies you've made some effort trying to see the listener and still can't see them, or it can mean that you can't meet someone in person.
Oh! It's not my native language either, so probably it makes sense to wait until a native speaker comes :)
Russian «видеть» in «Я тебя не вижу» is not used for understanding, only for getting visual information.
as a native english speaker (ages 7-30): the "see" in "I do not see you"/"I can't see you." is only regarding the visual. "see" only means understanding in a context where it is explained i.e. "I can see your meaning."/"I can see your point of view"/"I can see where you are coming from"(see note below) the only exception being the phrase "I see", a short version of "I see it" where "it" can be either visual or understanding depending on the context.
person A: Look! There's the explosion!
person B: I See! (i.e. I see it=the visual sight of the explosion)
person A: THAT is the meaning of life!
person B: I See! (i.e. I see it=meaning/understanding of life)
note: "I can see where you are coming from." is an idiomatic phrase meaning the same thing as "I can see your point of view", so the phrase is refering to the understanding and not the visual location of where the person "came from". (sigh) Unless the context involves someone who just arrived to your location.
Technically yes, but the default word order is to put the object pronoun before the verb (unlike objects expressed by nouns and phrases: these usually come after the verb).
When you move the word to change the word order like this, you add emphasis to the last word. So, in «Я не ви́жу тебя́», «тебя» is emphasised, i.e. 'I don’t see you', 'It's you whom I don’t see' (the important fact is that I can't see you and not someone else).
Now that's a good question! I think it's accusative since we don't generally use genitive for living beings (Я не вижу Светлану sounds much better than *Я не вижу Светланы).
Accusative of most* animate nouns looks like genitive.
Тебя is a pronoun so it doesn’t really follow the same rules as nouns, but in this cases, it is: both genitive and accusative are тебя.
The form *тебю doesn’t exist.
* Except most singular feminine nouns (мама — Acc. маму, Gen. мамы) and some masculine nouns that are declined like feminine (папа — Acc. папу, Gen. папы).