Ends up with a message saying it's blocked by Adobe.
FontSquirrel allows font publishers to request to be put on a blacklist.
Not only that sometimes if the font can be converted, it often looks like crap when viewed in a browser.
Sometimes automated conversion doesn't turn out perfectly and adjustments or alterations to the fonts are needed. (This is outside the scope of this site, and a potentially large topic.) The TypeKit blog has often published posts detailing some of the things they've had to do, if you're curious.
I assume that the generator is primarily for people to convert their own fonts, but if you purchase the use of a font then you can only use it for Web if the terms allows you to?
Correct. There's no universal answer to this. Fonts are generally licensed, not purchased outright. You need to refer to the license you, or the designer, agreed to. Some foundries don't allow embedding of any kind, some include it in the base license, others allow for the purchase of more liberal embedding privileges, etc.
(Strictly speaking, those designers probably shouldn't even be sharing those font files with you–again, refer license–but this is largely overlooked as a matter of practicality.)
The fonts used in PSD's are often Adobe Fonts, I can't find anything that suggests I can convert those and use them on the web. So I'm wondering if anyone knows the legal rights around using Photoshop supplied fonts on the web?
There's no such concept as "Photoshop-supplied fonts" for purposes of your question; it doesn't mean anything. As to Adobe's terms, here's the page with their various documents. I can tell you it's highly unlikely you're allowed to do your own conversion. That said, and particularly because of, they now own TypeKit, and have made a good chunk of their library available through it. Plans are affordable and there's even a free level, though it does require a fairly unintrusive badge.
Some other foundries don't have any relationship with an embedding service and so their license may specify that you can do your own conversion/embedding(eg. FontSquirrel) as long as it meets certain technical criteria, and so on. Again, you'll need to refer to their own documentation for such details.