If I were to chose, I'd say profiles are better than signatures. They're less pushy and less likely to be harmful to your site's rankings.

What is the primary benefit you're looking to get?  Algorithmic benefit? Direct traffic? Branding?

More often than not, signatures and links from profiles are nofollowed so you probably won't get any link juice from either.  If I were to chose, I'd say profiles are better than signatures because it's better to have one link coming from a stronger page than a bunch of links with the same anchor text coming from a bunch of weaker pages.  If you have a page of content on your site that would be important to your target audience, you'd be better off not using either of those methods, as they're on the spammy side.  You'd be better of promoting through social media channels.

Any drops in traffic during those times?  How many times has it disappeared? If you made substantial changes to the pages that were showing up in the results (content, navigation, programming), that could possibly cause such a thing.

Becca,

That wouldn't be too strange if it's very new and since you say that you're building it, I assume that it is. If it's older than a couple of months, it would be strange.  Getting some links and some social media activity going for it would probably work to stabilize things for you.

It's a toss up but I think you should redirect it to the internal page. The relevancy will be better and it's a good opportunity to get a strong link to an internal page. You could have a small blub on that page that welcomes visitors who were redirected to lessen any disorientation from it. Yes, you may lose a bit of branding but those deep links can be hard to get and I think it would be worth it.

If you delete the page (URL) with the link (clickable words that takes the visitor to another URL), Google will eventually recognize that the URL no longer exists and drop it from the index. That may take anywhere from a few days to a month or more.

Once the URL is out of the index, whatever links were on that URL won't count against you.

You may use the URL removal tool in Google webmaster tools to try to speed up the process of removing the URL from the index.  There is no need to use the disavow tool for a URL that resolves to a 404 page.

If the links are coming from URLs that no longer exist, then think  of them kind of like ghosts.  Their likenesses are still showing up on reports to give you a scare but their bodies aren't around and can't do you any harm.  Eventually they'll fade away from the reports any they'll be nothing more a distant memory. So, no, non-existent links don't need to be disallowed.

Sorry, I was watching a ghost movie last night.