Does John Petrucci pay income taxes to the government?
Yes. Any money he receives is income, whether it is wages and salary from his company, or other income. All income is taxed by the IRS and state unless it is specifically exempt.
Does Would the government recognize Petrucci's membership in Dream Theater as an "official" occupation?
Not sure what you mean. The government doesn't care about "official occupations" for most purposes, other than, say, unemployment insurance, workers' comp., and other employee benefits issues that the government regulates. Or are you again talking about taxes? Most taxes are income related, not employment related. But in either case, yes, he is being paid a salary as an employee of his corporation, so yes.
Does being a touring musician make you "legally employed"?
Depends whether you are drawing an income. In John's case, yes. He is employed by his corporation.
If Dream Theater broke up tomorrow and its members had to find new jobs, could they put their membership in the band as previous job experience?
Sure. Whether it is relevant experience is a different matter, but a lot of that is all in how you describe it. He has been a corporate executive for over 20 yeard now, for example, and has been involved in hiring and firing of employees, negotiating contracts, marketing, payroll and employee benefits, managing corporate finances, etc.