I'm in Los Angeles and I've got my own creative website with my short films and photography -- should I add that link to my Black List profile?

yes, although i have a feeling most industry pros don't really look at profiles.

Should I list other screenplay competitions that my other features have gotten into?

unless you were a nicoll finalist, the blcklst 8 will likely be your most impressive accomplishment. definitely don't list semi finalist at vail film festival, which nobody's ever heard of (sorry).

How can I make it abundantly clear that I want a manager without sounding desperate?

trust me, whoever you're querying/pitching to already knows you're looking for a rep.

I've gotten reads, then polite no's from managers earlier this year on a 6-scoring feature I wrote. Would it be worth following up with them and mentioning the 8?

if the 6 was the same script as this 8? probably not. they already gave you a chance and passed, there's nothing really that can be done to change their mind. they've got 500 others scripts to get through.

if this 8 script is new and different from the 6, sure, query away. but don't mention the old 6 script.