Looks like you may be an example of the *extremely* rare phenomenon where one sibling got a completely different "deal" of genes than the other(s). If all your siblings are sisters and no brothers, that might explain the "no Y-DNA match" - women simply don't have a Y chromosome to receive or pass on. (On the other hand, many autosomal tests just don't look for Y chromosomes or mitochondrial information, so that's another possibility.)
The reason Y- DNA works for distant line testing is that it is passed down *on the Y chromosome only*, and is *never* involved in any remixing *at any time* - the only way it can change at all is by internal mutation (which is sometimes used to bracket approximately where one male line branched off from the parent stock, if there are enough male descendants to test). But the mutation rate averages somewhere around one step per century.
mt(mitochondrial)DNA is passed down on the maternal X chromosome *only* and is never remixed either. It has an even slower mutation rate (hardly at all in a millennium), which makes it useful for detecting remote ancestry but not so much for the kind of bracketing that's possible with Y-DNA.
Due to the method of transmission, neither Y nor mtDNA can "skip a generation". On the other hand, any break in same gender transmission will "lose" that information (e.g. all daughters and no sons means no Y-DNA passed on, while all sons and no daughters mans no mtDNA passed on). Otherwise, yes they can be passed on with little change for centuries and even millennia. A particular flavor of the Y-DNA haplotype R1b (R-M269) is associated with Niall of the Nine Hostages (fl. circa 4th century AD), but molecular studies indicated that it didn't originate with him - there was already a small(?) kin-group carrying the same haplotype.
To date various peerage and royal lineages in Europe have been investigated and identified - most of them, including the Bourbons of France, have turned out to be some variation of R-M269, which is very common all across Western Europe. (The Bourbons were a headscratcher and a disappointment to some theorists who believed, partly on the basis of some previous tests which turned out to be flawed, that the French royal haplotype "should" be G. On the other hand, the first of the Bourbon kings, Henri IV, was only ninth cousin to the last of the Valois kings, so the Capet-Valois question remains open.)
The Stuarts not only were another flavor of R-M269, there were enough volunteering descendants to start bracketing which line stemmed from whom. (It's not common to have that many available and willing.)
Discovery and identification of the remains of Richard III under that car park raised hopes of identifying the "Plantagenet Y-DNA" type - but since that depends on confirmation from living collaterals (Richard had no surviving children), and since the Tudors had done a wholesale job of eliminating male and even some female Plantagenet claimants, those hopes were dashed. (Richard turned out to be haplotype G, four Somersets were the ubiquitous R1b, and the fifth was an I.) Who *wasn't* whose "baby daddy" is a popular exercise in speculation - there's really only one possible "iffy" link on Richard's side (his paternal grandfather), while the Somerset side is rife with them, starting with John of Gaunt himself (who was rumored in his own time to have been illegitimate, a rumor which seriously annoyed him all his life) and continuing down the line.
Unfortunately there is zero chance of acquiring any ancestral DNA from John of Gaunt - his tomb and everything in it burned up in the Great Fire of London in 1666. Next-best bet would be the notorious "Princes in the Tower" (Richard's nephews), but the Crown has nixed any further investigation of those remains. Other options remain contingent on *finding* male Plantagenet remains in good enough condition - which hasn't happened yet.
A while back there were some hopes of detecting the Y-DNA haplotype of the old Dukes of Normandy and thus of William the Conqueror (which wouldn't help with the Plantagenets, as he was *not* a direct male-line ancestor), but the investigation was foiled because at some unknown time someone had played shell-and-pea with the Ducal remains, and the body in the sarcophagus wasn't who it was supposed to be.
We're not *sure* what the actual "royal Tudor" haplotype was (no lineal male descendants, no available ancestral DNA), but it is likely to have been yet another version of R-M269, as most (admittedly distant) collaterals have turned out to be that type.
To date no male Hanoverian descendants have volunteered for testing, so we don't know about them either.
