I know it's a mistake in principle to keep referring back to English, rather than just letting Welsh be Welsh, but in this respect I don't honestly feel that it's that different from English usage. (I say this having had a similar problem with Old English, where the form of the definite article changes, and I had great fun trying to figure out which of the two nouns the 'the' went with, until I realized that it was basically the same as Modern English...)
Would you be happy with the following definitions?
A princess is the daughter of a queen.
The princess is the daughter of the queen.
I think that in English the 'daughter' bit is effectively unmarked as to definiteness: what matters is whether there's one particular, specific queen or not. (Similarly, "I bought a sheep's head from the butcher" = "the head of a sheep"; the sheep has -- had! -- only one head, so it gets a 'the' in the second version, but in the first version it's the fact that it's not a specific sheep that matters. If it's a specific sheep, "he saw the sheep's head poking above the bars of the stall" = "the head of the sheep".)
So in Welsh, Mi welais i ben dyn / Mi welais i ben y dyn (I think that works, although I've randomly changed it to a man's head rather than a sheep's...)
And then in the case of Anna's house, what's going to decide whether you need a 'the' or not isn't the house, it's whether or not you need a 'the' in front of 'Anna': "Anna's house" = "the house of Anna", "the Anna's house" = "the house of the Anna".
(I hope I'm not labouring the point -- there are languages that would certainly do things differently. In Greek you'd need a definite article even with the name, and you would say "the house of the Anna! )