My thoughts for types:
Batman: ILI 5w6 sx/sp
Joker: Obviously Beta NF IMO; he seems like a 4w5 so/sx so IEI>EIE
Harvey Dent: LIE, or perhaps IEE, 6w7 so/sx (Nick gets a point for me being 6 because I actually identified somewhat with this guy )
Alfred: ESI (this is the most obvious typing of them all IMO); seems like a 2w1 sx/sp
Rachel Dawes: SEE; seems like a 7w6 sx/so
Lucius Fox: LSI; seems 1w2 sp/sx
Gordon: LSE or SLI; 8w9 or 9w8 sp/sx