Nginx 403 and 503 errors

[MatShepSEO]

I have a client with a website that is hosted on a shared webserver running on an Nginx server. When I started working on the website a few months ago I found the server was throwing 100s of 403s and 503s and at one point googlebot couldn't access robots.txt. Needless to say this didn't help rankings!

Now the web hosting company has partially resolved the errors by switching to a new server and I'm now just seeing intermittent spikes in Webmaster Tools of 30 to 70 403 ad 503 errors.

My questions:

1. Am I right in saying there should (pretty much) be no such errors (for pages that we make public and crawlable).
2. Having already asked the web hosting company to look in to this. Any advice on specifically what I should be asking them to look at on the server?
3. If this doesn't work out, does anyone having a recommendation for a reliable web hosting company in the U.S. for a lead generation website with over 20,000 pages and currently 500 to 1000 visits per day?

Thanks for the help Mozzers

[MatShepSEO]

More discussion of this on Google+ too: https://plus.google.com/114779041142136013466/posts/U5PTFcRtbdT