Learn English – Verb play vs Present participle playing

The exercise wants the student to transform "Her dress was poor" using a verb form of dress.

Answer: She dressed poorly.

Why is "She dressed poorly" a better answer than "She was dressed poorly" (or poorly dressed)?

First, all of these sentences are a little odd. We don't normally say "Her dress is poor", but we say "She is well/badly/neatly dressed". In "She is well/badly/neatly dressed", dressed is an adjective.

As for the other matter that you pointed out, that "dress seems like a causative verb", it's true that dress usually used in the meaning "to put on clothes" (either on yourself or on someone else). But, the verb dress is also used when we wear clothes in a particular style or for a particular occasion. The definitions 1.1 and 1.3 given by Oxford dictionary should make this point clear:

dress
1.1 [with adverbial] Wear clothes in a particular way or of a particular type:
'she’s nice-looking and dresses well'

1.3 Put on clothes appropriate for a formal occasion:
'we dressed for dinner every night'

So, the best answer is: She dressed poorly.

From your linked website:

Relative clauses can also be used to add subordinate information.

The two examples are of additional information, meaning that it would be fine to leave out this information, even if it is interesting. Linking with the present participle gives no more or less emphasis than using other kinds of linking.

I plan to go to the doctor tomorrow, making it the first time in a while I've been.

I plan to go to the doctor tomorrow, where I hope I get a clean checkup.

I plan to go to the doctor tomorrow, if I can get there with all the traffic.

With all of these, the second part is subordinate information, but all are of (more or less) equal emphasis. You can even put two (or more) in the same sentence:

I plan to go to the doctor tomorrow, making it the first time in a while I've been -- if I can get there with all the traffic.