Your Nitrox class will go over this, but the answer to your questions is maybe and yes.
The issue with have O2 cleaned gear is the presence of high concentrations of oxygen coming in contact with the gear. Now without going into some of the estoteric discussions of what that % of oxygen is (read CGA standards), the generally accepted and taught standard is oxygen concentrations of 40% of less can be largely treated the same as air. Therefore, subject to your manufacturer's recommendations, which I haven't read, generally regs produced in the last 10 years are nitrox ready to 40%.
The issue of your tanks is slightly more complicated. There are three primary ways of filling your tanks with nitrox. One of those ways, partial-pressure blending, begins by filling the tank with 100% O2 and then diluting it with air (very clean air). Because there is 100% O2 in the tank during this process, you would need an O2 cleaned tank. So the answer comes down to how you will get your nitrox. If you get your nitrox from different sources (e.g., boats, different shops, etc...) its best just to have a tank dedicated to Nitrox.