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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!

Exactly.  Assuming your directory site structure is the same, it will do a blanket redirect from one URL to the new one.

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.

Very simple.  Find your .htaccess file  for the URL you want to redirect FROM and add a line like the following:

Redirect 301 / http://www.example-site.com/

Just change out your own destination URL of course.  Don't ignore that solo slash, that's key as well.

In my experience it can take a while to see sitemaps updated, particularly if you are a small site and are updating it daily.  Unless your site really is changing everyday, I would submit it less often and/or just be patient.  You didn't indicate how long you've been waiting, but at least a few weeks to a month or two if your site is new or "small".  If there is a problem, your Webmaster Tools will let you know.

I agree with Crusader as well.  Sections should be broadly included in the sitemap...specific individual pages aren't really a concern with this (assuming you link well to them).  You are only meant to provide the structure of the site to Google (main section, sub sections, subdomains, etc.) and let them come index when they are ready.  But I do think you probably need to move onto other efforts and be patient for Google to get around to visiting your site.