To note - my mom found out about this site when she received a note telling her she needs to pay close to a $1000 to license the photo that was ripped off from her site. Otherwise she wouldn't even have found out about this.
To me this is smells like the real motive behind this. It's not about trashing your SEO, or stealing traffic, but rather to scare those small website owners into paying some kind of 'fee'.
It's pretty safe to assume lots of small websites for local businesses use various photos they found online or downloaded from Google images, rather than their own content or images they bothered paying license for. Copying those images onto a seemingly legitimate site and then filing a claim for license payment seems like a pretty good strategy to milk money out of innocent site owners.
I'd be curious to see the note your mum received and what kind of language or tactics it's using.
However, tricking Google is another matter. The site I used to work for has had probably hundreds of other sites steal content from it, some of them really egregiously (IE mirroring the entire site). It didn't matter though, because our site had a better reputation and had the content first. For example, let's compare your site to that other domain:
1) Your site was registered over 4 years ago. The knock-off site was registered 6 months ago.
2) Your site is hosted by a reputable web-host, linode. Their's isn't (or at least the whois for their IP range doesn't look reputable).
3) You had the content first.
4) Your site has some reputable incoming links, their doesn't.
5) Their site's IP is hosting tons of other sites, yours isn't (I'm assuming)
You are obviously the more reputable site, and that is why you are going to rank and they aren't.
The best defense is just to look more reputable than your competition --- the easiest way to do this is to be more reputable than them!
Three quicks things that I noticed you might want to improve:
1. Change the http://doremi-nj.com --> http://www.doremi-nj.com redirect from a 302 to a 301. A few incoming links are coming to the non-www version of your site, and theoretically a 301 is better for passing SEO juice than a 302. (In simple cases like this I'm sure Google figures it out anyways, but might as well help them).
2. Make sure you are giving Googlebot a sitemap with links to all your content to help them crawl it first. There isn't one in your robots.txt, so I'm not sure if you have one. You can just upload this directly into Google Webmaster Tools as well.
3. Have better meta descriptions. For example for the search "guitar lessons livingston nj", you come up 2nd. Very good! But, the snippet for your page is "Do-Re-Mi School of Music & the Arts93A South Livingston Avenue,Livingston, NJ973-758-1500 We offer instruction in Acoustic and Electric Guitar and Bass, ...". This is not a good snippet. For your main 20-30 landing pages (which you can find in GA) write custom meta descriptions that are going to look good in the Google snippet. This will not only directly help your traffic, but this is also is a good reputation indication! Human written meta descriptions look more human than robotic ones.
The #1 thing Google is concerned about for search is webspam. The #1 thing they are concerned with for Adsense is click fraud. So in this situation, we probably have a) a site which gets little traffic because google search has detected it is webspam and b) a site which makes little money because it gets little traffic, and c) doesn't engage in click fraud. Therefore, it still is a member of Adsense, but gets no search traffic in google search.
http://www.theguardian.com/money/2013/apr/15/passport-applic...
Any suitably popular site is going to have all sorts of people ripping off your content. Google knows this, and work very hard to deal with it. I just searched for a bunch of things related to your site and the legitimate pages appears to be winning them all, and I don't even see the bad page showing up in the top 10.
If the site is sapping traffic, file a DCMA against their hosting provider (though there are some shady web-hosts who won't do anything about it). Obviously if they are filing a DCMA request against you, defend yourself. But just because somebody is ripping off your content doesn't necessarily mean you need to do anything.