What does Mike mean by his question, and what is Jimmy's response?

Mike Dolan is asking if Jimmy has another safecracking job lined up. This is intended to show that the two men know each other pretty well, but it is also intended to show that Jimmy doesn't trust even his best friends. He answers:

“I don't understand. I'm representing the New York Amalgamated Short Snap Biscuit Cracker and Frazzled Wheat Company.”

This is one of the drawbacks of being a criminal. Furthermore, O. Henry wishes to show that Jimmy is notorious as a professional burglar, the best in the racket. Jimmy must realize that his reputation is not doing him any good but could do him a lot of harm. The warden knows he is a career criminal. Mike Dolan knows it too. And worst of all, Ben Price knows a lot about Jimmy's modus operandi. When three safecracking jobs are pulled off a week after Jimmy's release from prison, Ben Price knows that Jimmy is back at work.

“That's Dandy Jim Valentine's autograph. He's resumed business. Look at that combination knob—jerked out as easy as pulling up a radish in wet weather. He's got the only clamps that can do it. And look how clean those tumblers were punched out! Jimmy never has to drill but one hole. Yes, I guess I want Mr. Valentine. He'll do his bit next time without any short-time or clemency foolishness.”

Jimmy is smart enough to realize that things are closing in on him. He must have been thinking of going straight even before he saw Annabel Adams and fell in love at first sight. She merely triggered a psychological change that had been evolving under the surface ever since Jimmy got arrested for the "Springfield job" and received a four-year sentence. He only served ten months of that sentence before getting "sprung" by influential friends, but that in itself was sobering because he had only expected to be in "stir" for three months at most. 

The fact that Mike Dolan runs a bar suggests that a lot of men know about Jimmy and talk about him. Bars were male hangouts in O. Henry's day. No doubt Jimmy was a common subject of conversation. Jimmy probably knew that if he told Mike anything about his plans in confidence, it wouldn't be long before everybody heard about them. It would be bad enough that Mike was sure to tell some of his regulars that Jimmy Valentine was back in circulation. They would all be wondering what he was going to do next. Jimmy goes to far-away Elmore, Arkansas in order to create a new identity as Ralph Spencer. But he finds he can't escape from himself.