You're going to laugh at me, but despite all my Zippos and peanut lighters and titanium EDC whatsists, my current favorite lighter that I'm rocking every day at the moment is one of these. Not the Bob Marley pot leaf eight ball skull and crossbones variety. I just have a plain neon green one. But altogether it's only a shade over 3mm thick at its widest point which is up at the top where the fill port is. The body is wedge shaped and would almost taper down to an ice scraper edge at the tail end if it weren't rounded over. It's made to be slipped into a traditional flip top cardboard cigarette pack but you can slip it into just about anything, including the otherwise useless business card holder in many folios, or a credit card slot in your wallet.
I can't help you for a strap-on low profile cigarette case. I just use a plain single layer metal one.