How do I get my updates button (next to installed) to show?
IIRC closing the updated clears the data. It will automatically reappear the next time a resource is updated/added.
How do I use Help functions?
When Logos 6 (or whichever version actually introduced this feature) came out and I saw the "Explain the Screen" link (when you click on the question mark/help icon), I assumed that the idea was to provide contextualized help. Instead, one gets a . . . description of the logos desktop.. . .
However, F1 invoked from a specific type of window does invoke contextualized help. If you are in notes, the help file will open on note documents. If in a passage guide, it will go to "working with guides".
How do I get the point marker in the Atlas to not overlay the place name?
Problem:

Solution:
click on the cross, circled below, to close the search and the map will remain open but the orange point will be removed.

Can I have different default Bibles for the Old and New Testaments?
You can do this through Advanced Prioritization in Library. See the wiki article https://wiki.logos.com/Prioritizing#Advanced_Prioritization
How can I download media to watch offline?

In the media window that opens when you click on a video from the course (1) click on the little arrow and (2) just click download all media. I am not sure how you get at the MP3s but I would look for a similar option on the android/ipad mobile app and see what happens when you download all media with these.
How do I find what Biblical cities are in modern day Iraq?
I searched:
"today's iraq" OR "modern iraq" OR "modern day iraq"
That pulled about 100 refs, which I'd then use Logos to find where, using map search syntax.
How do I find as many cognates for a given word as possible?
run a morph search "root:διδωμι AND root:λεγω" (note that I not even needed to type Greek characters, I could write root:g:didomi and then root:g:lego).

Maybe you could provide some more details on your request, so that those fellow users who read Hebrew can help you with the right suggestions. It might be that e.g. the HALOT lexicon already lists what you need, or that you could run a search or build a concordance on the Hebrew Bible.
Not answering the OP's question, but fascinatingly related:
https://www.amazon.com/Dialect-Geography-Syria-Palestine-1000-586-B-C/dp/0812279271
Just looking at the cover map is tasty. Plus there's a TOC and the surprise-me is the index. Dr Heiser's (sp?) inscription volume is more direct, but not as fun.
your Hebrew-English lexicon can help you. Brown, Driver, Briggs arranges all its entries by word roots, thus grouping words sharing common roots for you. The Hebrew and Aramaic Lexicon of the Old Testament and the Dictionary of Classical Hebrew both have lists of words sharing a common root at the end of many entries.
In Factbook what is the difference between the sparkline and the popup?
Problem:
[quote]

Answer:
Factbook is answering: "Where is 'Eagle owl' tagged on the lemma יַנְשׁוּף?" (answer: 2 occurrences)
The popup is answering: "How is יַנְשׁוּף translated across the entire Bible?" (answer: "great owl" and "owl" in LEB; "owl" in NRSV; "screech owl" and "owl" in NASB95; etc.)