I don't bother blocking cookies because there's honestly nothing they can put in them that I'm all that concerned about (sites only have access to their own data or any data you specifically give to them -- so an ad site's cookies probably only contain data about what ads they've shown you, which is a moot point if you're blocking the ads anyway).

Quote from here explains what cookies do:

"Cookies do not act maliciously on computer systems. They are merely text files that can be deleted at any time - they are not plug ins nor are they programs. Cookies cannot be used to spread viruses and they cannot access your hard drive. This does not mean that cookies are not relevant to a user's privacy and anonymity on the Internet. Cookies cannot read your hard drive to find out information about you; however, any personal information that you give to a Web site, including credit card information, will most likely be stored in a cookie unless you have turned off the cookie feature in your browser. In only this way are cookies a threat to privacy. The cookie will only contain information that you freely provide to a Web site."

But yeah, blocking cookies would be the reason why sites keep "kicking out" your passwords -- most sites these days use cookies to maintain your login, so if you block cookies, you're going to find yourself getting logged out at each new page. For this reason, blocking cookies is usually more hassle than it's worth.

For what it's worth, you can see what AdBlock Plus is blocking (and allow certain things through selectively) by clicking on the ABP logo (the stop sign). But it's not what's blocking your cookies, so it's not going to help with your particular issue.

If you really insist on blocking cookies, some of the info on this page might help to make it a more tolerable experience. In particular, look at the part about setting up exceptions, as you can set it to allow certain sites of your choosing to store cookies, but block all others.