Is it a good idea to use nofollow when a page has several identical links all going to the same place? I'm talking about links to my own pages, not external links. Say if there are 3 text links on a page, all pointing to page1.html, and also some graphic buttons or images pointing to the same page1.html - should I 'nofollow' all of them except one?
This is exactly why I don't like the nofollow attribute. With three links on a page pointing to the same URL, there's no issue. Yet you're worried, and understandably so, because Matt Cutts made mention of rel="nofollow" in his blog and, unfortunately, it has led to misinterpretation. rel="nofollow" was meant for external links. That way, you're not assigning any "link juice" to pages that don't deserve it, and you're also not inadvertently assisting spammy/banned content. I wouldn't bother with it myself. It hasn't affected me or my clients in any way, shape, or form so far.
I believe rel="nofollow" can also be used to optimize internal linking structure. After all this forum uses it, and I guess DP know what they are doing.