Questions of Protocol

I have a question concerning the proper way to follow-up on a submission. I had a manuscript critique with an editor from a closed house roughly five months ago. She was interested in the novel, but it wasn’t finished yet. At the end of the critique she gave me her business card. A month later, I sent her a picture book via the conference coupon. It is now one week shy of four months and I haven’t heard anything yet. This is a relatively short amount of time I know, but I am anxious to send it on to other publishers but feel this was an exclusive submission as I didn’t note otherwise. Which would be more professional in this situation, send a follow-up e-mail (which was on her card) or snail mail? I would walk a mile of hot coals to work with her, so I don’t want to do anything that comes across too casual. Any advice?

It's not an exclusive unless you tell someone it's an exclusive. Keep submitting. And you may email the editor at this point and remind her that she's had your submission for four months. Mentioning how excited you would be to work with her isn't a bad idea, either.

Is it OK for a writer (newly unagented) to approach one (or more) of the editors at a closed house who looked at another manuscript when she was agented?

If the editor in question specifically asked to see other work from that author (and not just other projects from that agent) in her correspondence, then yes. Be sure to remind her of that, and be calm if she wants to stand on her company's no-unagented-work policy. But it's ok to ask.

If yes, then can said writer use the email address she obtained via notes from the agent or should she snail mail to the address of the publisher? If this is not appropriate behavior ever ever ever, please discuss why not.

No. While this does vary some from editor to editor, many editors do not want anything in their email inboxes that they could reasonably handle via paper. This is not because we have a vendetta against trees, but because our email inboxes are monstrous already. They are not the place for slush. Agents get to email me because the stuff they send me is often time-sensitive. Email is for things that need a quick turn around. The author who emails a manuscript to me seems to be saying I am important rather than the preferable I have manners.