Sadly, none of these suggested four lines (purely male or female) appear to have present-day carriers. And in some parts of genealogies, links are doubtful, wherefore if DNA is found from another buried, identified person, it rather tests such dubious link than the identity of these subjects.
Generally, only fairly close relatives of these subjects can be named, to be reasonably certain of the genealogy linking such one to one of these subjects. The remoter a relative is, and remembering the pitfalls of medieval genealogies, the bigger is the risk that the underlying genealogy (either female-line or male-line) would be incorrect and the DNA test meaningless.
Group A
Y-chromosomal DNA is more difficult to extract from ancient cadavers than mt-DNA. Thus this could be impossible even if corpses are found.
Birger Jarl's well-attested legitimate male line went extinct in 1387 at the death of Olav IV of Norway and Denmark, pretender of Sweden. Perhaps corpses in that lineage are preserved in places where their identity is well known: king Haakon VI, young king Eric XII, king Magnus Ericson, dukes Eric and Valdemar, king Birger, king Valdemar, king Magnus Ladulas...
Also tose several guys who were lagman of Ostrogothia, as well as Birger Brosa, his sons and grandsons and the pretender Knut Katarinason. And the Hama (Ama) lords.
Birger Jarl's illegitimate son's male line (House of Arnö) went extinct at some stage in late 15th century. If a corpse of one of those noblemen can be found, such test would be helpful, provided the Y-DNA can be extracted in sufficient amount.
As can be seen from the tree linked below, last putative male lines of the House of Bjellbo went extinct in late 16th century. If a corpse of one of those (for example, Tillbakaseende Ulv) can be found, such comparison with Birger and Erik would actually test something about whether Tillbakaseende Ulv really were male-line descendants of the House of Bjellbo, and of earl Folke the Fat.
http://genealogics.org/descend.php?personID=I00079667&tree=LEO&displayoption=male&generations=6 For finding Bjellbo dynasty's possible hidden bastard male-line descendants, y-DNA markers of Birger Jarl and duke Eric should be published, and presently living males, when having a DNA test, could have theirs compared. It is not impossible (though certainly not assured) that in some places, even in Sweden, unknownly agnatic Bjellbo descendants, presumably of illegitimate lines, live yet.
Group B
We know nothing about Ingrid Ylva's mother's identity. And we do not know anything certain of whether Ingrid had daughters. Or sisters. Genealogical reconstruction has proposed a sister to Birger Jarl - a lady from whom Tyrgils Knutson and those Hafrid families would descend. If such lady was Birger's sister, she could as well have been Magnus Minnesköld's daughter with another woman than Ingrid.
In any case, fairly likely those female lines went extinct, as far as genealogy can be historically followed, within a handful of generations.
One could set wishes that mt-DNA of Tyrgils Knutson, of Bengt Hafridson, of Magnus Kasi, of Bengt Uddson of Vinstorp, of Anders Israelsson [And], of Gustav Arvidsson of Vik, of Erengisle of Orkney, of Kettil Puke, of Erik 'rebel' Puke, of Ingeborg and Kristina Ulvsdotter (half-sisters of lord of Tofta), and of Kettil Glysing, if any of those corpses are known, be tested and compared with Birger jarl's mt-DNA. Such tests actually are testing more the hypotheses of descents from a reconstrued sister of Birger, than affirming the identity of Birger himself.
http://genealogics.org/descend.php?personID=I00079664&tree=LEO&displayoption=female&generations=6 Group C
Mechtild of Holstein's matriline seems to descend from Adelheid of Arnsberg.
Her female-line descendants are listed in the following tree. All they, as well as their sons, are those who would be carriers of mt-DNA to be compared with Mechtild's.
If corpses are found, probably sufficient mt-DNA can be extracted from some bone or suchlike, to make testing possible.
http://genealogics.org/descend.php?personID=I00050067&tree=LEO&displayoption=female&generations=6 Group D
It is fairly certain that duke Erik Birgerson's matriline emanates from Rikissa of Poland, wife of Magnus I and Sverker I.
However, it is possible that his matriline stems yet farther from a certain Berta of Este, in Italy, who may have been Rikissa's matrilineal ancestress.
Rikissa's female-line descendancy seems to have gone extinct at some stage in 15th century. Quite long time ago, for finding attestedly certain corpses to test.
Other female-line descents from Berta also seem to have gone extinct in 15th century.
Ladies listed in the following tree, as well as their sons, are those who would be carriers of mt-DNA to be compared with Erik's.
Corpses of sons are more likely to be known today than corpses of females. If corpses are found, probably sufficient mt-DNA can be extracted from some bone or suchlike, to make that testing possible. This matriline, mt-DNA lineage, includes plenty of important people, across Europe, and would be most interesting to be tested.
http://genealogics.org/descend.php?personID=I00020856&tree=LEO&displayoption=female&generations=6