Welcome - this is my first answer here too!
The fact that the sentence uses were + past participle should suggest that it's actually not using the past continuous. It's the past perfect. "When we were done, we ate dinner" means that at some point in the past, we ate dinner. Before that, even farther in the past, is when we were done with something. "We were done playing, then we ate dinner" demonstrates the order of events.
Playing, in this context, is acting as a gerund. This means it acts like a noun, and in this particular sentence is the direct object of do. It's saying what we finished doing.
In the second example, there's no to be verb because it's using the simple past tense with the verb died, instead of the past continuous. Giving birth to me is a descriptive phrase that explains how the mother died. It's using the present participle as an adjective.
Keep in mind that that's what the present participle is for; it's to act like an adjective. "She was speaking" has the same sentence structure as "She was smart" -- speaking and smart are both acting as adjectives.