Is "This sentence is true" true or false (or both); is it a proposition? From what I understand, a proposition is either true or false, but not both. "This sentence is false" can be neither true nor false and is thus not a proposition.

However, is "This sentence is true" true or false (or both)? And hence, is "This sentence is true" a proposition?
The dichotomy sentence/proposition is quite complex to manage, due to its philosophical implications.

See e.g. Nik Weaver,Truth and Assertibility, World Scientific PC (2015), page 4:


  Many philosophers consider truth to be fundamentally an attribute
  not of sentences but of some more abstract correlate of sentences called
  “propositions”. The idea is that sentences function by referring to or expressing abstract propositions, and it is these propositions which are the
  “primary bearers of truth”. This seems to be a common opinion, but it is
  controversial, with some dissenters denying that there even are such things
  as propositions.


Thus, if we want to stay in the realm of propositional logic, we can say that the basic entities are sentences, i.e. linguistic entities, that have a definite truth value.

If so, a sentence like: "This sentence is false", that can be neither true nor false, is not a meaningful sentence to be used in the context of propositional logic.

What about:


  "This sentence is true" ?


Is it paradoxical ? I think so.

Assume that the sentence is true; then its negation: "This sentence is not true" must be false.

But the negated sentence is equivalent to  "This sentence is false".

But if "This sentence is false" is false, then the sentence (asserting something about a sentence, i.e. a linguistic entity) "agrees" with the way the things are, and this means that it is true.

Again, we have reached a contradiction.
"This sentence is true" - call this sentence TT - is not paradoxical. It can be assigned either of the truthvalues T or F. The negation of TT is not the paradoxical Liar sentence "This sentence is false", since the 'this' in these two sentences refer to different sentences. The negation of TT might be rendered as "The sentence 'This sentence is true' is false" - call this sentence NT - which is awkward but quite self-consistent, since TT can indeed consistently be false; in fact, NT can also be true or false: as one would expect, its truthvalue must be the opposite of the truthvalue of TT. 
It might be worth emphasizing that being a sentence which can be true or false is nothing particularly remarkable. Most sentences of logic are in this category. None of this has anything at all to do with the double-slit experiment, Quantum theory or the nature of physical reality.