Writing my packet for performance review

I'll be actively pushing for a promotion

Will this performance review summary document be a promotion packet or a normal review where you would want to get an exceed rating to solidify your promotion case in near future review cycles?

Does your company have publicly available document examples for some performance reviews or promotion packets? If so, the document can provide answer to many of your questions. To answer your questions:

1. It depends upon your company or more specifically the org you are working in. Generally speaking, the review document should have details about your most notable projects with focus on the impact. Just to give you an idea, LinkedIn has 1500 characters limit on promotion packet. and no limit on normal performance review.
2. In general, I have seen that covering all the competencies is not required in performance reviews as long as you landed good impact on your projects. I have seen managers and reviewers focusing on holistic view of the candidate's performance instead of matching each point with a certain competency.
3. I would suggest including both: strong ones at the top followed by the good ones. Once you note down the achievements, you can compare them with the competencies required. If you see a ~80% fulfillment, you should be good to go. Otherwise better to reframe a couple of achievements or you can consider adding any minor achievement. Overall, quality is the key.

If you have good working relationship with some of your peers at +1 or +2 level, they might be open to help here. Having a glimpse at their document may help reveal the expected length of the document and in general the number of achievements to mention.

For example, I had a review document with relatively bad wording and was lengthy. However, I realized only when one of the engineers at +1 level showed me his document's first page as a glimpse.

How long should a self-review generally be?

Does your company not have guidance here? At Facebook, they were pretty prescriptive about word count, I believe the recommended 1500 words? I'd follow your company guidance and also run it by your manager.

Should I cover all or as many of these values and competencies as I can, or should I focus on just a few of them?

The way I'd normally structure my self-review would be a narrative at the top (what is the thing you want people to remember), followed by data to support each specific dimension that your company/manager cares about. More here: Making The BEST Case For Yourself In Performance Review: The Self Review