Interchangeably? Perhaps not 100% "interchangeably." But either article could be used. The resulting implication is largely the same no matter which article is used, although how that conclusion is reached changes slightly depending on which article is used.
When it comes to rights, I may have several rights, such as:
- choosing my own religion
- voting my conscious or my interests in an election
- driving a car
- buying alcohol
I have a right to be angry about this.
implies that I may have several rights (such as the four I just mentioned), but, right now, I'm talking about a different right: the right to be angry about some matter.
I have the right to be angry about this.
means that I am justified in my anger, but it makes no implied acknowledgement about any other rights I may have. On the other hand, it does NOT imply that I don't have any other rights, so one could still use it.
When talking about rights, asking which article ought to be used can be tricky. We have legal rights, moral rights, ethical rights, parental rights, and personal rights; it largely depends on the context. I might say:
"I have the right to eat a cookie for dessert."
but what does that mean? And is it even true? Most people have the legal right to buy a cookie and eat it, but there could be religious convictions, personal reasons, parental restrictions, or dietary restrictions to prevent them from doing so: it might break a declared fast for Lent, violate a diet plan or lifestyle choice, break a family rule ("No dessert if you don't eat all your vegetables!"), harm the health of a diabetic, or cause an allergic reaction.
For your four examples, I have no problem with using either article, and I'm hard-pressed to explain a significant difference between the two, other than the trivial matter I've already gone over. I think you could even mix-and-match them:
I have the right to be angry with you and a right to know the truth!
(I can imagine a suspicious spouse saying that with no small amount of ire.)
For the last one, there are two ways that it could be interpreted:
I have a/the right to protest against the government.
- I have a/the legal right to protest against the government (because such protests are explicitly protected under the national constitution); or,
- I have a morally justifiable reason to protest against the government (that is, even if the right to protest is unauthorized, I cannot in good conscious cower in silence)
I'm trying to figure out whether or not one article tends to map better to one meaning, or if either article could be used to express both possible meanings. I'm inclined to believe it's the latter case, that is, either article could be used to discuss both an established legal right, and a gross grievance.
I realize I have rambled a bit here, but there are several contexts in which we can talk about rights, and I felt it was important to at least consider a few of them before reaching a conclusion. My bottom line answer is that, in some contexts, one article may sound a bit more natural than the other, but, for the most part, either one could be used, and they will pretty much mean the same thing.