Hi John,
First and foremost, you want to know for your own peace of mind that pages from your site are crawlable and indexed.
The first thing I would do is identify your most important "money" pages and go add them to the index via Webmaster Tools in both Google and Bing using their "Fetch as Googlebot" and "Fetch as Bingbot" tools. In both cases, a clean fetch of those pages will tell you that the pages are accessible and crawlable. In addition to that, once a URL has been successfully fetched, both allow you to request that the URL be submitted to the index (and also linked URLs). There are limits on the number of URLs per month that you can submit, so make sure you pick the most important ones.
Perhaps another way to approach this is to go back to your analytics and see if you are actually getting any traffic from those terms that your Boss is obsessing on. While everyone wants to rank, the fact is that with personalized and localized results, it is quite common for a site to appear in SERPs from time to time and actually bring traffic for terms that it "doesn't rank for". If you are actually getting some traffic for those terms from a search engine, you have actual proof that your site has been served up in a result for that term...from that point you know that your job is improving how frequently that happens.
While you probably wouldn't want to say this to your Boss, responsibility for your rankings rests largely with him/her also. If they want to play with the "Big Boys", then they have to step up their game in the offline world as well. They should be asking you what the company can do to make the website more visible, not the other way round.
You need to address the question mark over the ecommerce platform. Again, for your own peace of mind (and given that there are some minor issues with nearly every CMS/Shopping Cart "out of the box") I would dive into any forum, available documentation etc that you can find on the platform you are using.
Use the Pro-tool to crawl the site and check that it is not creating duplicates etc. Find out about every known issue, find & present solutions, get your techs to write a fix if necessary. Make sure that you know when software updates are available, and that you are kept in the loop whenever changes are made that you have not initiated...KNOW that your ecommerce platform is NOT the problem and make sure that your Boss knows you can say that because you did the work.
Personally, I would also take a good shot at getting on the "front foot" by developing a solid marketing strategy that includes both online and offline activities. Basically, you need to show your Boss that you are in control here...you see the problems, understand why they exist and have a strategy for dealing with them.
Lastly, remember that as a SEOmoz Pro Member you can ask one Private Question in Q&A every 30 days. With all the negative attention you are getting, it might be a good thing to ask if a Staff Member or Associate can review the site and see if they can pinpoint any particular issues with the ecommerce platform, site content or optimization strategy. This will either get you some great help in fixing any issues you may not have seen, or just give you a confidence boost to help deal with the heat you are taking.
As for the endlessly forthcoming "expert" who found a tool to play with, you are probably going to have to find a way to put that unhelpful input into a proper perspective. Just remember, if you put up with the constant whining of the mosquito for long enough, sometimes you can actually manage to swat it right out of the air - and it feels GOOD! 
Hope that helps,
Sha
