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PR is based on links to a given page. Links from pages with high PR have more clout than those with lower PR. But in the mix pages with just a few links to other pages will pass on more PR than pages that are a long list of links.
Does anything else affect PR?
Placement in serp for a given key word is influenced by:
1)PR of the page
2)key word in title, H tags, and general keyword density
3)key word in the anchor tags of incoming links
What else? I know there is a lot more but it all seems so uncertain with so many opinions on what matters and what doesn't.
What I'm also trying to sort out is which changes I make will have to wait for the deep crawl and update to show and which changes can be put into play by the fresh bot.
It seems to me that keyword density, title and other on page factors can bring about changes in the serps right away and we have to wait until the deep crawl and ensuing update to see how link changes have influenced both the PR and the serps.
Perhaps I am incorrect in connecting PR with the deep crawl and other factors with the fresh bot.
Does anything else affect PR?
Only incoming links to a page, determine the PR of that page. From that you can conclude that the PR of your site across all the pages is determined by the link structure on your site.
Nothing else affects PR (well, of course a PR0-penalty would, but I guess that's not what you were asking).
It seems to me that keyword density, title and other on page factors can bring about changes in the serps right away
I don't understand that part. PR is calculated roughly once a month during the update.
Your pages in the SERP will usually only be updated at the same time, so onpage-factors don't help you "right-away" as soon as you implement them. Of course the exception is made by the Freshbot. If your page(s) get freshbotted, they appear shortly (and only for a short time) in the SERPs and then of course will onpage-factors make a difference.
But maybe I just didn't understand what you wanted to know...