Etymology Online suggests that the use of impertinent to mean "rudely bold" is connected to similar use in French, and happened around 1680. It also offers the tantalizing tidbit

especially by Molière, from notion of meddling with what is beyond one's proper sphere.

Now the first thing I think of when I see "Molière" is intermediate French class and the play Le Bourgeois Gentilhomme. Jean-Batiste Poquelin aka Molière was a French playwright whose satiric plays centered around characters who were taken out of their 'natural' element. In Le Bourgeois Gentilhomme, for instance, is the story of a middle class man who aspires to the aristocracy. He is like a fish out of water as he tries to learn to be a gentleman, and much hilarity ensues.

If we look closely at the etymology of the word, it is not too much of a stretch to equate "not pertaining to something" with "not a part of something." The association with Molière's works certainly could have helped impertinence take on this slightly altered meaning.

So it would seem that the "intrusive or presumptuous" meaning of impertinence is fashionably French.

The words relevant and pertinent are members of a group of words that—according to Webster's Dictionary of Synonyms (1942)—also includes germane, material, apposite, applicable, and apropos of. That dictionary distinguishes between the two words you're interested in as follows:

That is relevant which has any traceable connection, especially logical connection, with the thing under consideration and which has significance in any degree for those who are engaged in such consideration [examples omitted]. ... That is pertinent which is so decisively or significantly relevant that it touches the real point at issue or contributes materially to the understanding of what is under discussion or to the solution of that which is in question. [Examples omitted.]

I was surprised at the claim that pertinent indicates a stronger degree of relevance than the unadorned word relevant does, since I have always imagined that the two words were used for the most part interchangeably, though perhaps with pertinent more closely bound to "questions," and relevant to "issues." But when I checked the synonym discussion under relevant in Merriam-Webster's Eleventh Collegiate Dictionary (2003), I found that the same distinctions that MW made in 1942 were on offer (albeit in shortened form) sixty years later:

RELEVANT implies a traceable, significant, logical connection {found material relevant to her case}. ... PERTINENT stresses a clear and decisive relevance {a pertinent observation}.

So that's the steadfast view of Merriam-Webster. I remain skeptical about the level of real-world adherence to these distinctions in popular usage today.