Firstly, I have thought about this too before. Actually, only one word is needed for all of I, you, this, there, tomorrow, etc. You can analyse these as I, (the person I am referring to), (the thing I am referring to at/by/on ME), (the place I am referring to not at/by/on ME), (the day which is one day later than the day I am in). This goes for most words like those.
- Not that I know of. But let's create one for demonstration purposes: Loj-ban mi. It is Lojban, but there's a ban on the word mi (and vi, and ti, and other words incorporating mi as said above).
It would be able to express concepts like 'Bananas grow on trees', as this doesn't depend on the concept I. However, saying stuff like My name is Santa Claus., It's going well. and What are we doing tomorrow? would be impossible.
We could replace it with other concepts, for example, 'the speaker/writer'. But this is ambiguous, take this sentence: I don't know who said that, but the speaker is in big trouble when I find out!. (And, pulling it off correctly, it nevertheless depends on context). Names can be used, so the answer is actually yes.
Yes, as we proved above with Loj-ban mi, it is possible. Would it be learnable? Probably. But not usable in practice.