The culture - name issue is really only relevant with the change to arab culture I believe. Most european cultures use variations of the same names, so I have no problem when a culture shift appears and Henri's son names himself Heinrich, Henry, Enrique or something of the sort. Now if he starts calling himself Mustafa-something I do have a problem as the only way that would make sense would be if he adopted muslim religion first. I think the culture shift to arab is in general a problem as it also determines the looks of the character unlike any other cultural shift (well some pagans I believe use the same graphics).
And yes, I also use dynastic names for my characters, renaming any sons, nephews and grandsons according to a list of acceptable names (I use one of two rules, eitehr first born is named after father or named after grand-father, then the names of uncles in the paternal line, an ocasional uncle in the maternal line (if he marks hi time), a famous king of that generation etc., occasionally I also name characters for claims (playing Lancaser in the 1066 scenario I used the names Harold, Godwin as well as a number of other Godwinson earl's names to demonstrate my claim on the trone of England, of course that was in addition to Leofricson earl names and of course the all important Edward...)). I occasionally also change a character's culture, but only if I feel a change unrealistic (an occitan duke of Flanders just doesn't fit my view of realism when none of my lands lie in the south and I have only a single connection to an occitan family (I give more leeway to female names where I usually don't bother changeing them)) and of course I always change the name and culture of any of my dynasty members who turned arab by accident.
P.S.: Culture shift was quite common in the middle ages. Just among the Luxembourgish dynasty the female name Ermesinde is probably the second most famous (after Kunigonde), yet that name was imported from the Pyrénées (probably via Poitiers). The dynasty also changed from all German (rhenanian-mosellan frankish) to french (different branches changed at different times) largely due to marriages and to a lesser degree due to simple geographic proximity.
P.P.S.: And the example of Henri - Heinrich is also taken from the Luxembourgish example as the 1066 starting characters should really be Konrad, Heinrich (not yet born, heck his mother is an inbred child no one wants to marry in that scenario), Wilhelm (idem to Heinrich) and Ermesinde (whoever researched this even got her name wrong, in addition to her also being the inbred child of Poitiers future daughter...). The culture shift simply hadn't occured yet (and only did during either the Namur dynasty or during the switch from the Namur and Limburg dynasties).