Responses to your questions below.
Does anybody have any advice around how such situations typically play out?
Left untouched, these situations typically play out with one person growing more frustrated with people not listening, while others grow more concerned about whether or not the person is reliable and can do their job, until at some point the person quits. So, to break this pattern, change the outcome, and improve the overall situation, my advice is solve the immediate problem by addressing whatever questions or concerns you have, and then solve the larger problem by focusing on building a better working relationship with your immediate supervisors.
When I talk about addressing whatever questions or concerns you have, I refer to making sure you have what you need to successfully accomplish whatever tasks you are responsible for. Knowing what's going on in your boss's life may ease any fears or concerns you have, but maybe it's personal and they want some privacy? On the other hand, it's reasonable to know who you should give status reports to or who you should go to when you need help. And if you need immediate help, I'd present a list of what you need and ask for assistance.
In terms of building a better working relationship with supervisors, you can take steps to reassure people that you are reliable and take responsibility for whatever you have been assigned. They will be better listeners when they aren't worrying as much.
After people are listening, present the list of things you could use help on.
I would also encourage you to learn more about differences in personality and use this to guide your approach in working with people. This would include knowing your own personality as well as knowing the personality of people that you are working with. Check out https://www.16personalities.com to learn about different personality types and to take a test to identify your own personality. You can also Google "working with different personalities" for a lot of articles that might help.
I can give you an example of how understanding your personality can help improve your working relationship with others. I tend to be extroverted, which means I get excited and energized when working with others. Sometimes I overwhelm people as a result, and sometimes I'm busy thinking out loud by talking, when I really should be fully present and focused on listening to the other person. Understanding this about myself allows me to adjust my behavior so that I can adapt and work more collaboratively and more effectively with others.
If I were introverted, the situation would be different, because I would find that engaging with others requires a lot of energy. It can also cause a lot of stress and anxiety if I am expected to think out loud and make quick decisions - introverts like to step back and quietly think things through before making a decision, maybe even do some research in the process. If I were an introvert, understanding all of this would again allow me to adjust my behavior. For example, it might help me feel more comfortable being assertive and telling people, "That's a good question, let me check on something and get back to you with an answer".
In short: can I take time off for work-related stress even though some of my stress happens to be related to personal issues?
I would say yes, but it's really up to your supervisor. First make sure taking time off isn't going to cause any problems at work. And when you ask, keep it as simple as possible by just saying you would like some time off to relax and get caught up on a few things.
And do I have to explain my reasons to my not-boss or can I claim that its due to work-related stress and leave it at that without being more specific?
I hope you don't have to get specific, but you might be asked for details if you taking time off causes problems - mainly because your supervisor will have to come up with a way to prioritize between your needs and the needs of the organization.