I would similarly object if someone were to breach my paywall. If you don't like the NYT's, write your own newspaper, or use one of the many competing options.
However, I don't get why they're sending the payment to the blog's email address and not the NYT (disclaimer: don't know how the NYT blogs work). It seems to me Kachingle would be better applied at the site level. Given my limited understanding, I can definitely see how applying it at this granular level would annoy NYT - their paywall solution would make money for NYT first (and trickle down later) whereas this seems to make money for the bloggers first, and not for NYT. At the very least, payment should be split between NYT and the specific blogger. (And if this is incorrect and the payment is in fact going to NYT - then why on earth do they send it to so many different places?!)
I like the idea of Kachingle for reading across multiple sites, but the NYT blog situation just seems weird. No matter what though - assuming my evaluation is even correct - figuring this stuff out took way too much digging and reading on Kachingle's site.
They don't have to send the money to anyone. They could keep it for themselves. But if they did no one would use the service as it wouldn't make sense. Instead they donate to the writers personally. It's the same idea as United Way collecting donations and then forwarding those donations to various causes, while keeping a bit for overhead.
Not illegal. If they were hosting content, copying content or circumventing, then it would be a problem.
Their use of official photos and logos of the blogs without permission is not fair use though, that part will have to go. But they can certainly collect donations for a third party according to any system they want without needing the third party's permission.
So they are breaching a paywall? That wasn't made clear in the post. I assumed this was distributing funds for public facing content.