"I'll have ..., no ... " is fairly common, even though I'd personally still tack a "please" on at the end. ;) The "without" is just as common, particularly with people who don't feel rushed. As for the coffee I guess that there's a lot of local variation, but here people typically specify coffees by names like "flat white", "long black", "short black", "espresso", "macchiato" and so forth. With a "coffee, no milk" you'd probably get some strange looks. Almost like going into a craft-beer pub and asking for "a beer". ;)

This is a 'lazy' / 'modern' manner in which to phrase the request. (but more common than the use of the full sentence) The meaning is the same regardless of the context not confined to the ordering of consumables. E.g. There are four of us here today therefore we could play a game with teams of two. So we can quickly identify teammates two of us can play without shirts OR two of us can play 'no shirts'