The fact that folks link to you with a space in between and without and other variations is a good thing. That doesn't affect how Google views your brand. In fact it helps you cause it allows you to have a link profile that looks natural with lots of variations around the term trophy etc.
If you search for your brand name with spaces or without you can see that Google clearly recognizes your brand since they show your site first and with a decent set of sitelinks below your homepage link in the SERPS. If they didn't "understand" your brand name they wouldn't do that.
One thing that could help is that on each of your pages of the site I noticed you don't use your brand name on the page title. It is common to do something like this: " Sports Medals and Awards - Trophy Central" While that isn't the greatest title ever it is there to quickly illustrate that on nearly every URL if you have your branding in the title at the end or beginning whichever is best for you, that gives another signal to the engines as to what your brand is.
In looking at a sampling of 500 URL's for your site only 5 of them come close to meeting the standard length of 70 characters or so that Google shows in the serps. It is a tedious job, but handcrafting non keyword stuffed titles can help greatly in influencing what people are clicking on when your site shows in Google.
As for not showing for search queries that is a bit puzzling. I looked at your backlink profiles and didn't see any glaring problems there. I looked at the code on your site and while very bloated it wasn't anything un crawlable. You have 25,300 pages indexed in Google. That appears to be quite a bit especially since you have just over 7,000 pages listed in your .XML Sitemap, so you may have some duplicate content issues to resolve and your URL's are not very clean but again not a glaring reason to be not showing at all. I notice you have Google Webmaster Tools code on the site, have you received any notifications from them recently? Have you logged into GWT to check things out there?
Your product descriptions are quite thin as well. Which overall isn't good but again probably wouldn't have affected rankings overnight. When did you first notice the drop? I would fix as much as you can on the site like page titles, urls, dup content and give Google a little bit of time to see those changes and that should help. Hopefully someone else on here can take a look and offer some more advise about why the drop might have happened. 1st blush looks like a number of things all piling up and the penguin update might have pushed you down. Take care.