No problem Dan.
There is definitely no harm in ticking off the errors as "fixed" and then seeing which ones return. If they return I'm betting 1 of 2 things. 1.) That you still have a live link somewhere on the new site pointing to the page. The easiest way to test this is click on one of the pages in WMT and hit the "linked from" button. That shows you where google is getting sent to the page from. The second option is that the pages have an "external" link pointing too them. If that's the case you want to be double sure to redirect it so you get the link credit!
I'm not 100% positive on why MOZ is not showing the errors as quickly, but MOZ does a "deep" crawl of the web and then does a lot of computing. WMT is literally just crawling your site, and spitting out anything it finds to your account. A much easier job for WMT, hence it is pretty quick to report site errors too you. The errors in your MOZ account will most likely update when MOZ pushes their next mozscape index update (this will also be useful for you because MOZ will have better data about "external" links than WMT)
From the error you sent me it looks like you are going through a move of a site from .aspx to .php or .html. When you change languages like this you can end up with a lot of gnarly 404 pages, especially from search pages that have variables in the URL. If you have a large amount of these errors, so many that you feel you are getting a temporary "penalty" for too many site errors you can write a quick snippet of code in the .httpdconf file on your apache server to fix these old useless pages. (if your not on an apache server this can still be achieved, the code would just be different.)
make sure to put this snippet below all of your other individual page redirects
#redirect all old dead .aspx pages to the new homepage
RedirectMatch 301 /(.*).aspx /
You should still make your best effort to redirect each old page to its corresponding "new" page. However at a certain point if you have thousands to redirect, and many are just pages with with search parameters in the URL... there comes a point where you want to just hit all the remaining pages in 1 fell swoop.
I'm not certain on how long it would take the 301 to fully "pass" page rank in Google's eyes. I think it is safe to assume that once the old page no longer has a 404 link, and it is not showing up any longer in Google searches then the PR has been passed. MOZ page authority should reflect it once MOZ pics up the change, but MOZ authority is just very "similar" to PR. Its not an exact science, MOZ should pic up the change easily though, it will just take them a little longer time.