Water hammer is caused by moving water moving things when it stops abruptly (like when you shut off a fast acting valve - some toilet valves, ice makers, washing machines, etc. contain fast closing valves that typically cause water hammer). So, it doesn't sound like that's your problem. I will say that some brands of shower valves do call for arrestors to be installed.
It sounds like you have a pressure balanced anti-scald valve and that there's some air in the lines. Normally, any air in the supply lines gets purged once you open a valve, and then isn't an issue any more. If you're on a well, you may have an issue with the pump injecting air or even some gas from the well itself. If you have a new water heater, sometimes the anode rod will react with the water to produce gas, and that is relieved when you use some hot water. In that case, you'd need to replace the anode with a different metal one that won't react (or treat your water).
Do you get a spurt of air when you open any other hot water valves in the house?
A pressure balanced shower valve requires the inlet pressure to be fairly even between the hot and cold supplies. If there's air on one side, it can slam the pressure balance spool valve and shut off both supplies until they become balanced again. The valve is just trying to do its job, and if it is really air, you need to address that first, then things should return to normal.