Do you believe this to be true?

I'm Jewish. In the Orthodox Jewish world people live their lives according to halakha -- traditional Jewish law -- and resolve business disputes as well as matters like divorce before batei din -- Jewish courts. I don't see people panicking about that and I see no reason to panic about Muslims following sharia or having sharia courts either.
I suppose people who do sincerely worry about this have in mind the kind of application of sharia that we hear about from Afghanistan or Saudi Arabia: brutal flogging, cutting off people's hands, executions. Guess what? Similar stuff is part of halakha also. But these corporal punishments and executions do not in fact happen among Jews, partly for reasons internal to halakha itself, partly because the law of the land does not allow them, and partly because nobody in the community wants that kind of punishment to exist in practice. I am certain that the many Muslims I know here in Canada do not want these sorts of punishments being carried out either, nor would Canadian law allow them.
So yes, I do believe that quotation.

I agree, but it is a "religious based law" because it's rooted in a "holy book," the same way Mosaic Law is rooted in an ancient "holy text."
I understand your point, but I don't think we need to defend Sharia Law, we just need to point out to all who hear the fear mongering that no one in this country needs to fear Muslim America trying to implement Sharia Law because the US Constitution forbids it. To implement Sharia Law is to establish a religion in this country, and the Constitution is clear on the matter: "Congress shall make no law respecting an establishment of religion........."

You can argue that is is neither, or that it is both.

I think that from the point of view of the actual legal systems in most countries Sharia is, to some extent and in principle, a system for arbitration. The problem is that the parties to it are not all of equal power.

Nope.

In theory, maybe. In practice, rarely.

It is all that as long as it's not given any legal authority

Well, honor killings, fgm, all good right?