First, my right and my ability to exercise it are two different things. My being too poor also doesn’t impinge on my right to own a gun, but it certainly affects my ability to do so. Moreover, you are focusing on the example and not the law itself. I can as easily exercise my right to bear arms by finding a big stick or a suitable rock. Further, exercising my right doesn’t require the government to restrict it in order for it to exist. And, of course, violation of my rights would generally require the government to be involved.

As far as speech is concerned, yes, I’m exercising my right to free speech if I say anything in a public space, including asking you if you’ve seen my lost cat. It likely wasn’t going to be infringed in that particular example. This still doesn’t allow me to go on private property and ask people about my cat.

Rights don’t have to be at threat to be rights. A right is “something that one may properly claim as due.” Note that there is nothing there which requires it to be threatened.