When you have a multi-part question where there is more than one component or UI element involved, is it better to ask a single question where everything is neatly presented in context but the answers could focus on one sub-component or should the question be broken up so that each sub-component is a distinct question and a link is provided to the other question(s) for context?
It depends, but generally it's better to simplify your question so that you describe one thing which can be answered. You will get better answers that way. If you need to provide background to help explain the problem you're having, you can do so, but even then, try to keep it to a minimum.
If you really do have several questions that need to be answered for you to be able to continue, consider asking them separately and then linking to the others from within the question body. That way people can look at the others if they want to but can also choose to just focus on each individual component question.
Generally I've found that when I think I'm going to ask a question that has many sub questions, really I just need to know one thing and I haven't figured out how to best formulate it yet. If that happens, we can help you in the comments, by helping edit your question, or you can visit meta to ask about it.
If you're not concise, you risk losing your audience.
If the readers don't agree with part 2 (or 3) of your question, you risk losing your audience.
If the question is too specific, not generalizable or interesting, you risk losing your audience.
Even if you do all these things, you may not get a (good) response. Doing these things will generally remove barriers to participation, not promote commenting.
Like all UX discussions, clarity is the foundation of good interaction.