Hi Berner,
"So when you are looking at your sites most popular keywords, do you make the most popular keyword your homepage?"
Technically yes, but be smart about it. One reason why the puppies query is so much more popular is not just because people want information about the puppies or want to adopt them. You're literally getting searches in there from people who just want to see cute pictures and videos of Bernese Mountain Dogs (because who wouldn't?!). If you are interested in this traffic, go for it. If you are more focused on people who may convert into customers (whatever that means for your particular website), pick home page title tags and headings that better reflect the home page's goals.
I am not sure what you mean in this instance by "more link juice". "Link juice" is a colloquial term for PageRank, or the authority that flows into and around a website from inbound links.
"When you have keywords that have parts of other keywords, is it safe to use the longer keyword because it has parts of the other phrases?"
Yes - this is pretty common. If I have a site where I sell accessories for swimming, I can have a page about "Speedo competitive racing suits" that covers the term "racing suits" and likely also "competitive suits" and to a degree, competitive swim suit terms related to the Speedo brand. I will have mentioned "swim" or "swimming" on the page, so Google is aware of what type of suits I sell.
For question three, bolding and italicising terms is absolutely not necessary anymore. It's questionable when this last worked, but I'm guessing it was before I started working in SEO in 2006 Placing your primary keywords in URLs and as H1 tags, however, is common and expected. It would be odd to have a page about Bernese Mountain Dog grooming that did not use that phrase as its H1 / main heading, and it wouldn't be at all unusual for that page's URL to be example.com/bernese-mountain-dog-grooming. The H2s, if there are any, should be used for sub-headings throughout the page. So to take the grooming example, perhaps there are there H2 tags (you can have more than one) regarding the best brushes, coat clipping it applicable and shampoo.
I hope this makes sense, let me know if you have any issues.
Best,
Jane