Lighthouse (Strollos) Ices

Question

I don't have the ingredients on me, and I'll get them if you need, but are Strollos Ices (at the famed Lighthouse in Deal kosher? does it matter that it has become something of an issue in that many many many people still eat there even though they think it isn't kosher. The vast majority of people are NOT eating there b/c they have seen the ingredients and believe it is kosher...They just don't mind eating non-kosher (in the same vein as bringing your own cheese to a non-kosher pizza store) In light of this, might it be mariit ayin to eat there since the vast majority of people do believe its not kosher, and by someone they know who keeps kosher eating there might just assume its kosher but for the wrong reasons? Does that matter? I ask b/c as much as kashrut is a black and white, its either kosher, or not, I do believe its also a mindset in that if your were to eat something that just HAPPENS to be kosher, although for all you knew or cared it was not kosher, that would be just as bad as eating something non-kosher. And conversely, if you happen to eat somehting non-kosher that you had every reason to believe was kosher, then you ate kosher. I guess what I'm asking is how much does intent play a role.... One more question...Would it be possible to publish a list of commonly found ingredients that are NOT kosher? In other words, what are the few chemicals/derivatives that are not kosher, are there places where we could get tripped up, etc...? This would seem easier and shorter than a list of all the kosher ingredients.... Thanks Rabbi...A million thanks and all the hakarat hatov for this INVALUABLE service that helps to truly EDUCATE all of us.