Patrick is right, the migration guide is an excellent source to make sure you've covered all the bases.  That said, setting up 301 redirects is gonna have to happen.  It may hurt, it may take a lot of time, but it's the best way I've found to minimize any negative impacts to rankings and traffic.  There are, of course, ways to speed this up, but I wouldn't suggest looking for a work-around.  It sucks to do but that's why new sites and new content on new domains can be such a nightmare.

It's mind-numbing as a task but your manager/client will be very impressed by it.  Make sure to look over the inbound links to your site to ensure no pages are missed thereby breaking some of your link network.

Definitely do not worry about the disavowing nofollows, unless the domain itself is somehow egregiously offensive to your brand.  Otherwise, you're safe to stick with dofollows.

I've never encountered an official limit to image filenames, and I'm not sure there is any SEO impact (other than just the engines ignoring alot of the filename in their crawl).  Putting the product description in the filename seems VERY unnecessary and I would try to get some rationale around that.  But for the web usage overall, I've never encountered a filename that is too long.  50-70 characters is alot but not prohibitively.

For paid search, I dont feel there is any real negative impact to using a separate domain for your landing pages.  If this were an SEO question, I'd advise against it.  But for PPC, it's safe.  I don't think there is any better approach if you are dealing with an offense "penalty" like that.

On a side note, make sure to the end user you are clear and above board and not using any bait-n-switch tactics with this.  Use your logo, your general site design, and so forth on the landing pages (which, I assume, link back to your main domain).  It's bad form to do otherwise and while the engines won't penalize you, your customers will.  PPC e-commerce is very conversion-focused so make sure you dont inhibit that by making the customer flow feel spammy or unethical.

But there is no real risk to using other domains for PPC landing pages.  Many large corporations do that currently.

If the redirect to the testing site actually is temporary, this is one of the very few instances I would recommend using a 302 redirect.  This way, the engines will understand that you aren't making this a permanent move and will, hopefully soon, remove the redirect when the staging site goes live.

All my clients are impatient with Google's crawl.  I think the speed of life on the web has spoiled them.  Assuming your site isn't a huge e-commerce or subject-matter site...you will get crawled but not right away.  Smaller, newer sites take time.

Take any concern and put it towards link building to the new site so Google's crawlers find it faster (via their seed list).  Get it up on DMOZ, get that Twitter account going, post videos to Youtube, etc.  Get some juicy high-PR inbound links and that could help speed up the indexing.  Good luck!

Like Mike said above, there still isn't enough info provided for us to give you a very clear response, but I think he is right to point out that you shouldnt really care about the extinct pages in Google's index.  They should, at some point, expire.

You can specify particular URLs to remove in GWT, or your robots.txt file, but that doesn't seem the best option for you. My recommendation is to just prepare the new site in the new location, upload a good clean sitemap.xml to GWT, and let them adjust.  If you have much of the same content as well, Google will know due to the page creation date which is the newer and more appropriate site.  Hate to say "trust the engines" but in this case, you should.

You may also consider a rel="author" tag in your new site to help Google prioritize the new site.  But really the best thing is a new site on a new domain, a nice sitemap.xml, and patience.

You would need access to the domain to set up the 301.  If you no longer can edit files on the old domain, then your best bet is to update Webmaster Tools with the new site info and a sitemap.xml and wait for their caches to expire and update.

Somebody can correct me on this if I'm wrong, but getting so many 404s and 500's already has probably impacted your rankings so significantly, that you may be best served to approach the whole effort as a new site.  Again, without more data, I'm left making educated guesses here.  And if you aren't tracking your rankings (as you asked how much it is impacting...you should be able to see), then I would let go of the old site completely and build search traffic fresh on the new domain.  You'd probably generate better results in the long term by jettisoning a defunct site with so many errors.

I confess, without being able to dig into the site analytics and traffic data, I can't give direct tactical advice.  However, the above is what I would certainly do.  Resubmitting a fresh sitemap.xml to GWT and deleting all the info to the old site in there is probably your best option.  I defer to anyone with better advice.  What a tough position you are in!

Like others have said, a 301 redirect and updating Webmaster Tools should be most of what you need to do.  You didn't say if you still have access to the old domain (where the pages are still being crawled) or if you get a 404, 503, or some other error when navigating to those pages.  What are you seeing or can you provide a sample URL?  That may help eliminate some possibilities.