Length of url & number of folders have some importance (see also http://moz.com/blog/15-seo-best-practices-for-structuring-urls - point 6 & 11) but I don't think they are major ranking factors.

What is important however is the depth of the site (how many clicks needed to reach the content you are integrating) - As you are integrating a old site on in another domain - I would try to make sure that you have sufficient crosslinks between the part you are integrating & the existing content. If you only have 1 link from the home to the "integrated" site, you'll be moving the old content one step deeper in the structure, which might have a considerable impact on your ranking (I was unfortunate enough to experience it on one of my sites)

If you check your page with external tools you'll see that the general status of the page is 200- however there are different elements which generate a 4xx error (your logo generates a 408 error - same for the shopping cart) - for more details you could check this http://www.webpagetest.org/result/151019_29_14E6/1/details/.

Remember that Moz bot is quite sensitive for errors -while browsers, Googlebot & Screaming Frog will accept errors on page, Moz bot stops in case of doubt.

You might want to check the 4xx errors & correct them - normally Moz bot should be able to crawl your site once these errors are corrected. More info on 406 errors can be found here. If you have access to your log files you could check in detail which elements are causing the problems when Mozbot is visiting your site.

Dirk

If you want to know how much traffic this domain is getting you could easily put a filter in Analytics which allows you to see the domain name when checking the pageviews.

Create a new view in Analytics and apply a filter with following settings:
Filter Type: Custom
Select: Advanced
Field A -> Extract A: Select: Hostname value: (.)
Field B -> Extract B: Select: Request URI value: (.)
Output To -> Constructor Select: Request URI value: /$A1$B1
Mark Field A: required & Mark "Override Output Field

You could also combine all the different copies in one set in the Search Console - which would make it easier to check if there is an impact when activating the 301's.

Regardless of the traffic, the 301 remains the best option - some people say it does dilute pagerank - Google says it doesn't

As both Search Console & Analytics are not retroactive you could already make them active, wait a few weeks in order to collect sufficient data and then activate the redirects.

Dirk

It's not really necessary to 301 these pages - a 403 status code informs Google that the access is denied (Literally: The server understood the request, but is refusing to fulfill it. Authorization will not help and the request SHOULD NOT be repeated.)

Normally these pages will disappear from WMT after a while. If you find these 403 annoying in your WMT reports, you can always 301 them - but this isn't strictly necessary.

Removal tool - Google's advice is not to use the tool "to clean up cruft, like old pages that 404" (source: https://support.google.com/webmasters/answer/1269119?hl=en).

rgds

Dirk

If the pdf's are in a separate folder on your site - you could mark that folder as noindex in robots.txt

As far as I know, it's not possible to add a noindex to a link.

rgds

Dirk

The main reason why errors are listed is that you can solve them (if necessary). If these are old pages that don't have existing links on your pages - you can just forget about these warnings. However, if these warnings appear because actual pages are linking to non-existing pages this will lead to a degraded user experience and user experience is a factor which counts for SEO.

If you look at the 403 errors - normally WMT lists how the bot got to these pages. If the pages that are linking to this 403 pages are still on your site, you have to remove these links.

If you have dropped in traffic, you could try to do a full crawl of your site using screaming frog of Xenu, to do a quick check-up of the technical health of your site.

If you still have an old sitemap, or the most popular pages in Google Analytics from the period before migration, you could also use these url's as input for Screamingfrog - and check if all pages were properly redirected. If errors pop-up, these would be the ones I would redirect. I understood from your initial question that the 403's where coming from very old pages which were never meant to be accessible.

rgds

Dirk

Probably the easiest way to deal with this is to chose one of these pages as the main page & use canonicals on the other versions to the main one.

So - suppose you would chose _/?show_products=all _as the main page - than page /?show_products=48 would have /?show_products=all as canonical.
As you also use pagination - the option of the canonical offers the advantage that you can put the /?show_products_all as canonical url on all the pages  (check also this page: https://support.google.com/webmasters/answer/1663744?hl=en)

Another option would be to put a nofollow on the other display options. So by default people would arrive on /?show_products=24 - the links on show 48 - all would be of type "nofollow'

You might want to check this article: http://googlewebmastercentral.blogspot.be/2014/02/faceted-navigation-best-and-5-of-worst.html - the scope of the article is a bit larger than the question you as here but some of the points are applicable.

Hope this helps;

Dirk