A fact is the object of a statement that is true.
Truth is a condition of propositions satisfied when utterance corresponds to what is. For example, "2 + 2 = 4" is a mathematically, or axiomatically true proposition. The truth of "all dividends require financing" is self-evident when you know the meanings of "dividend", "require" and "financing". "The Earth is roughly 93 million miles from the Sun" is empirically true and is falsifiable (you need only demonstrate that the earth is not roughly 93 million miles from the sun to show that the statement is false.) "I feel glad" is a statement of self-knowledge and true when uttered sincerely (Note that this is a case of what is "true to you" tho it is not empirically verifiable by anyone other than the particular self uttering the statement). In each instance of the preceding true statements, they follow the form of "P" iff P, or, the proposition "P" is true if and only if P is the case, i.e. the correspondence of utterance and what is.
Reality in the sense you describe is psychological (i.e. reality "to you"), not philosophical (what is). Reality simply is what is, i.e., reality (the world, the case, states of affairs) is that which is known, i.e. reality is that which is empirically verified and knowledge of reality is empirical verification of what is (else how do you know what is?) As you have asked this is a philosophy forum, it is worth pointing out that philosophy translates from the Greek through the Latin to love of wisdom. Love in the sense of original utterance was akin to virtue, respect or reverence. Wisdom has often been called the intelligent application of knowledge, but, intelligent according to whom? Wisdom, for all that is said of it, simply obtains knowledge. Without knowledge to be obtained, there is no philosophy to be had or done. Philosophy, to be redundant, is respect for obtaining empirical verification. For example, if you look at the etymology of psychology, you will see it derives from soul, spirit, breath and self. In the former two instances, the term is imponderable and no knowledge is to be obtained. Of the latter two, the first is in the domain of biology (specifically physiology) and the second is of interest to the philosophy of psychology inasmuch as it is worth pointing out that only you can empirically verify whether or not you feel glad when you state, "I feel glad".
As for your assertions, it is not that a fact cannot be disproved, it is that it can be falsified and it is verified. To disprove the fact that "all ravens are black" one need merely demonstrate a single instance of a non-black raven. To disprove the statement "it is raining" one need only look outside and see that it is, in fact, sunny.
Truth is not a matter of belief. It may be "true to you" that Angelina is better off without Brad, but it is true that Angelina has filed for divorce. It may be "true to you" that two plus two equals five, but if you stand two plus two feet from a train track and jump the distance as a train happens by and are thinking you've a foot to spare, you'll likely never set foot in a philosophy forum again ;) It might be true to you that the world is flat and if so likely any demonstration to the contrary will fall flat short of launching you into space and circumnavigating the Earth so you can empirically verify it's non-flatness with your own senses.
As for your conception of reality, in addition to distinguishing what is true from what is "true to [you; me; us; them]", also consider the difference between the object (e.g. the world) and the content (e.g. propositions about the world).
Hope that helps.