.pdf documents are crawled by bots and they accumulate pagerank just like .html pages.

You can include links in them to other documents on the web and bots will crawl those links and pagerank will flow through them.

.pdf documents can be given a "title tag" equivalent by opening their properties and giving the document a title.  This title will display in the SERPs.  .pfd documents can be hard to beat in the SERPs if they are optimized and have links from a competitive number of other web documents.

Lots of document formats behave this way.  Excel, PowerPoint, Word for example.

In my opinion, .pdf documents can trigger a Panda problem for your site if you have a lot of them with trivial or duplicate content (as in print versions of web documents).  They can be given rel=canonical through .htaccess to solve the Panda problem but Google often takes a long long time (sometimes months) to recognize the canonical and use that instruction.

Go back and rework our .pdfs so they at least contain a link back to the homepage?

Yes!  Absolutely!  And, link them to other relevant pages.  If  these are reference documents they could be pulling in a lot of links and traffic from other sites.

In addition toAs well as configure the hyperlinks so they open in the same window instead of a new one?

In my opinion, this is not an SEO issue.  This is a usability issue.   I would have them open in the same window so the back button is available.

I have hundreds of articles on my site and they range in length between 500 and 4000 words - with lots of photos.

My retail pages usually have several hundred words of unique relevant content and they often link to article content (500 to 4000 words) that explains how to use the product, how to select, how to repair, etc.  These are clusters of content around retail products.  In my opinion, it is the best way to rank.... but sometimes your articles will rank better than your retail  --  but, that's OK because we have house ads on those pages to let visitors know that we sell the stuff.

Compared to most others, my articles are usually long, ranging from 1000 to 4000 words.

I don't think that I would publish a single article that is 15,000 words unless there was no way to break it down, such as a long story.  However, if it is informative content, I would probably break it into at least three or four shorter articles that are stand-alone about a single topic.

The New York Times has a lot of really long articles (10,000 words plus).  I honestly don't like them because they take too long to read.  If they can't break them into multiple short articles then I think that they should simply shorten them.

I am currently reviewing my strategy when it comes to categories and tags on my site.

Everyone should do this.  I do it once every year or so.

Having been no-indexed for some time, and having many tags with just one entry I am thinking that this is not optimal for SEO purposes.

I agree.  I don't use tag pages.

Categories - Change these to Index, but only after adding a hundred words or so by way of introduction (see this example).

Where I have used categories, I wait until I have a substantive amount of material to appear on the category page.  I visited your sample page and can't tell if the hundred words or so at the top is yada yada yada content or real beef.  If it is real beef then go with it.  If it is yada yada yada then wait until you have a large enough number of posts that your page is of substantive length.

Also, I run periodic traffic assessments on my category pages.  If some of them are not bringing in the traffic or at least showing traffic growth then I delete them (301 redirect to the blog homepage).  My philosophy is that a compact site competes better for difficult, high-traffic terms if it does not have a lot of useless pages.

I am thinking of deleting all tags with just one entry,

Yes, these never should have been created.

and trying to merge those with just two or 3 entries where it makes sense

If this is going to create pages that compete with category pages then just delete them.

I will keep these as no-index, but I think this will mean more optimal distribution of link juice within the site.

In my opinion, tag pages are dangerous if they have snippets of the same content that appears on category pages and on the main blog page and its paginations.  Also, tag pages that are same topic as category pages are a bad idea in my opinion.

If you are not indexing pages they will pull zero traffic from search.  If you have links to them then you linkjuice is being scattered into potentially low-value pages.  I am all about internal linking but keep my philosophy that compact sites compete more strongly for the difficult queries where the big traffic is earned.

I edit my images frequently without seeing problems.    Just edit the image.  Don't change the size or the file name or the file type.  Sometimes they might disappear for a day or a few but are back in the rankings.