More generally and seriously, Azul likes a mean streak. He is an asshole with very amoral business dealings and he wants an SO that isn't going to be shocked and horrified or otherwise genuinely put off by that.
Intelligence is also important. Genius is appreciated but not required. He can handle some explaining but if he has to lead somebody by the hand like a toddler it's not going to work out.
Also he will never ever admit it but he needs somebody that's willing to push him around sometimes and set boundaries because he will almost always take the mile even when he shouldn't.
If it's company of positive CR public, or just in a space where he feels safe like LeBlanc, yes. Give him the PDA. Let him flirt and lean and steal kisses and generally be a menace.
Mix and match with any of our recent CR, with SFW
Jamil. Jamil is Azul's type, he doesn't even try to hide it, we all know Book 4 is just Azul writing love letters to Jamil.
That depends.
If it's genuine public public, he'd rather keep any PDA on the subtle side because he doesn't like drawing attention randomly.