Some Questions Regarding Agents

I know rejections are just that, regardless of how nicely they are phrased as a let down, but what about the way a book is rejected to an agent? Is that written any different? If you're not writing to the writer, are you more honest as to the reasons for rejection?

Sometimes.
The reason we're not always up front about our reasons for rejection is simply that we're not sure how you'll react. This is true of any agent we don't have a relationship with, either. If we have a sense of how the agent (or the author) will respond to constructive criticism, we're more likely to offer it. We'd much rather help people than upset them. Telling where the line between those things are, though, is very difficult when dealing with strangers.

EA, I wonder if all editors respond that way to all agents from the agencies you mentioned in your post. I once had an offer from a young agent at one of those three agencies, and I turned it down because the clients he referred me to had not yet made sales, and described him as submitting to one or two publishers at a time, then waiting many months for a reply. I chose an agent who is not in NYC and not with one of those agencies, but who had an impressive list of recent sales in Publishers Marketplace. Was that a mistake?

Nope. Sounds like you made the right choice. Good for you for doing your research! Having an agent at a high-powered agency is not necessarily the same thing as having a high-powered agent.

Since, to stand a chance, we have to query agents just as widely as we submit to publishers, and there's all this talk about an agent/client relationship being like a marriage (there'd better be a LOT of Mr./Ms. Rights out there), how do we avoid landing with a dud?

Do just what the person above did—ask for client contacts, and talk to those people about the treatment they've had and the results they've seen.

My agent works at a great agency (though not one of the ones you've named), but I'm beginning to wonder about her as an individual. My ms has been out to six houses for five months, and we've only heard back from two (both rejections). Obviously she's not a "push this to the top of the pile" name. Now I just hope she's not a "trash it" name!

Don't assume this. The "trash" list is very short. Most agents have no chance of getting themselves on it.
Your agent shouldn't be letting your manuscripts sit, though. She should be able to get you a response in less than three months, or be able to tell you why. Part of an agent's job is keeping track of which editors have which manuscripts, and for how long, and if it's been a while it's their job to politely and professionally nag the editors. And maybe subtly imply that the editor is in danger of hurting their relationship with the agent.