Sounds like you are in a lot better shape than marcuswhinus; you now KNOW what the problem is on your site.
I am confused, however, about you not being able to get “access to your site”. Can you not ssh or ftp to your user space? After all, that is where you will need to do your tuning, and optimizing and via the PhpMyAdmin interface to your MySQL host, so you can index your tables and such.
If you can’t ssh or ftp to your user space, you should generate a ticket immediately so you can begin to bring your site’s resource usage back into line. If you show that you are making the effort, I’ll bet you can get support to help you - maybe something as simple as enabling WP’s cache, doing something to curtail comment spam, or curtailing the use of some mis-behaving plug-in might just do the trick.
You seem to have a very good command of the English language, so you should have no trouble figuring that out per the Terms of Service to which you agreed when you signed up for an account - short answer: nothing.
Of course, if you are within the first 97 days of your service, and qualify regarding your method of payment, you can get all the money you have paid Dreamhost back. If, on the other hand, you are talking about “consequential damages” or “business interruption expense”, you are out of luck, as the ToS very clearly absolves DH of any such responsibility.
Just a note for the future: If you are going to have “business critical” sites that use the resources it appears you might be using (based on your report of DH’s statements to you referenced in your post), you might be well advised to make a business decision and spring for you own server so you don’t have to share resources.
All that said, if you can provide more details (WP version, cache status, Plug-ins used), and need help curtailing resource usage or need help in indexing your tables, I’m sure there are those here on the forum who are willing to help you with that should you need it.
Work with DH to address the problems; they will generally go the extra mile if they can.