Law professor here.

1. This contract does appear to apply to the data breach check site run by equifax. To be specific, the contract at http://www.equifax.com/terms/ describes its scope of coverage as "ALL OTHER WEBSITES OWNED AND OPERATED BY EQUIFAX AND ITS AFFILIATES" which would appear to include https://www.equifaxsecurity2017.com (which links those terms).

2. Arbitration clauses are very bad in cases like this, because by far the most effective technique for forcing companies to compensate people and deterring similar problems in the future is the class action, and this arbitration clause, like so many others, includes a waiver of class actions. Class actions are important because litigation is really expensive and class actions are the only generally applicable way to aggregate enough small claims to make them financially viable. (They're the way we keep corporations from stealing five dollars from everyone in America.)

3. As someone down-thread correctly noted, arbitration clauses are also extremely enforceable. The short version is that the federal arbitration act puts a very big thumb on the scale in favor of the enforceability of even really oppressive arbitration clauses, and the Supreme Court has added a couple of thumbs of its own.

4. I can't tell whether Equifax meant to do this. The cynical interpretation is that some evil person decided "ok, there's going to be a panic and everyone is going to go to our website to check if their data is breached, so let's sneak in a way to get all these people out of class actions." The less cynical interpretation is "someone threw up a website, and the standard procedure within the company for throwing up a website is to include a link to these boilerplate terms of service." No way from the outside to tell which of these stories is true, and I'm not sure it really matters.

5. There are arguments that a sharp lawyer could make to try to convince a court that this clause is unenforceable. I'm happy to go into them if people want, but, offhand, I would much rather not have agreed to such a contract than have to try to convince a court to bounce it.

6. Yes, there is an opt-out. The opt-out is only useful for people who have actually read and understood the contract, which even with the press coverage is likely to be a vanishingly tiny percentage of the people who have inadvertently "agreed" to it. So it doesn't make it meaningfully less evil.

7. If you want to learn more about this kind of issue, I recommend Boilerplate by Margaret Radin. http://press.princeton.edu/titles/9837.html She does a great job of explaining why this kind of thing is a complete disaster and also bears no relationship to our traditional conception of what a contract is or should be.

Editing to add:

8. I guess I'll add really briefly that the key argument that this clause doesn't even apply would be that the contract distinguishes between "product terms of use" and "site terms of use," and that arguably the arbitration clause only applies to the purchase of products and registration.
If checking breach status doesn't count as purchasing a product, maybe that part (with the arbitration clause) doesn't apply on the terms of the contract itself. HOWEVER, registering for some kind of identity protection service as a result of having checked breach status probably does, so that's cold comfort to a lot of people who might use the site. I'll have to parse the contract more slowly and carefully before having any confidence in much more along these lines.

(Standard warning: nothing here is individual legal advice, contact an attorney in your jurisdiction before deciding whether you'll use this site, I'm just speaking about the overall interpretation of this contract and what we should think of it as citizens.)