"This is the woman, whose son is sick."

Translation: Das ist die Frau, deren Sohn krank ist.

[Danmoller]

Does deren agree with Frau, not with Sohn???

[javax]

Yes, "deren" -> femenine, plural / "dessen" -> masculine, neutral.
Das ist die Frau, deren Sohn / Das ist der Mann, dessen Bruder / Das ist der Mann, dessen Schwester.

[hutcho66]

Correct.

[Druckles]

Does it matter what article one puts as a translation to 'This'? I.e. is 'Die ist die Frau' acceptable? If not, why not?

[hutcho66]

Generaly 'Das' is used if the person/animal/thing hasn't been introduced yet. So here, as the 'das' is before 'die Frau', it is the most common article to use. After the woman has been mentioned you would use 'sie' of course.

[kkulonja]

Can one use "Dies" (= this) instead of "Das" here?

[Haesselmaas]

Why "dessen" and not "wessen" and why couldn't I use "Dies ist"??

[Slydiad]

I think "wessen" is "whose" when you're asking a question, and "dessen"/"deren" are "whose" when you're making a normal statement.