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Not that I can think of directly, there are lots of creative methods that I've seen used (but I can't vouch that they work.) I personally wouldn't worry about passing a little juice off to an affiliate site, unless you are talking about several (or dozens, or hundreds) outbound links on the same page.

Your friend is right. If there are 10 links on a page, and 9 are marked nofollow, that 1 followed link is getting less juice than a page that has only 1 link on it.  The division happens regardless to whether it is nofollow or followed.  I believe I remember reading this on SearchEngineLand, or it may have actually been a Matt Cutts video.  If I can find a reference to cite, I'll edit my post/add another comment.

People don't mind "category" pages. You might find it better to have a button that says "Select Your Region" which leads to a page with the 36 links on it (and include a bit of text, so it's not just a page of links - good for users and search engines to develop relevance with).  This will help with how your PR gets spread across your deeper pages.

I'd take an invite if people are throwing them around

This is another great one. I've built plenty of professional relationships with previous clients (we refer business back and forth), as well as with a few graphic designers who work mostly in the SEM side of things (banner creation, tracking, tweaking to increase conversions).

Guest blogging. It's nothing new, it's nothing that requires a lot of talent, but my god! It does the trick for most clients I manage. Helps me acquires custom anchor text dofollow links from themed sites and the long term relationship with these blog owners are worth their "weight" in gold!  You get a great link, you get some referral traffic, and you expand your network. Triple win!