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My second job, atleast for the next few weeks,(just started a new second job, working out my notice), is a place that serves coffee. Ex has been coming in for every weekend morning since about late February. He never used to. When he comes in he literally doesn't say a word to me or anyone. I treat him as I do every customer. He pretty much throws his change down on the counter and turns around snickering and laughing. Do I just ignore it or say something? It is really awkward to have my supposed coparent act like that. At my new second job I work with the daughter of the friend he used to live with. Do I put up with it if he comes to my other job too?

Ignore. Smile sweetly and ignore. He is looking for a reaction. DO NOT GIVE IT TO HIM! Look up BIFF response and Grey rock. That helps me with my family members. They bait and I smile sweetly and grey rock.

My DH's ex NEVER acknowledges us or looks at us. Because we typically see her at a kid event, I always say hello and/or goodbye and use her name. Her behavior is her baggage but I want the kids to see that I treat their mom civilly.

As Hope essentially wrote, he's just jerking your chain, seeking a reaction.  Responding could feed his ego.  Just keep it business, no emotions nor guilting.  While you probably should be polite and shouldn't grimace or frown, whether you should smile depends on a variety of factors, the workplace, the background, whether other customers are present, etc.  If a coworker is available you could let the coworker serve him.  If he misbehaves then inform the management.

Based on his behavior elsewhere as mentioned in other posts, you may decide to proactively inform your manager that your Ex could create scenes but hasn't yet (there).