This does not require being a particularly brilliant chemist—nor does it require any trickery or riddling. All of these operations are possible, if not straightforward.
He told me that he can turn a radioactive metal into a gas!
As just one example, the radioactive metal uranium (U) can be converted into a gas. In fact, this is a normal step in the process of converting naturally-occurring uranium (which is mined) into nuclear fuel. Naturally-occurring uranium is mined from rocks and (through various means that are not important) converted into a solution. That solution is separated, filtered, and dried to produce concentrated uranium oxide, which is often referred to as "yellowcake", since it is a yellow powder. In order to turn this into the type of fuel that is used by nuclear reactors (U-235), it is "enriched", which increases the U-235 concentration from <1% to 3–5%. The enrichment process requires gaseous uranium, so the solid uranium oxide is converted to uranium hexafluoride, which is a gas at relatively low temperatures.
As another example, the radioactive metal thorium (Th) can be converted into one of various compounds, such as ThBr4, via bromination. These actinide tetrabromides occur in a gaseous state. See, for example, Hildenbrand et al., 1990 and Beeching et al., 2001.
[H]e can also turn a couple of gases into a couple of different gases very quickly.
Sure: hydrogen (H) can be fused into helium (He). This is called "nuclear fusion". The sun does it. Well, maybe this would require a pretty brilliant chemist to accomplish here on earth. Especially safely. :-)
How about transforming fluorine (F) gas into neon (Ne) gas by bombarding it with x-particles? That works, if you believe that Ben Rothersburger must be talking about elemental transformations.
If Ben isn't limiting himself to working with elemental gases for his transformation (and he doesn't claim to be anywhere that I can see), then that opens up a whole bunch of possibilities. Methane, for example, can be converted into carbon dioxide using some chemical catalysts—in fact, this is something chemists are looking into as a way to combat climate change. And there are various catalysts for converting CO2 into O2.
Without catalysts? Sure, that's possible, too. Combine gaseous HCl and F2 in a simple replacement reaction to get chlorine gas:
2HCl + F2 → 2HF + Cl2↑
He can even take the stuff from the matches and turn them into diamonds!
Well, yeah. Matches contain carbon, which can be made into diamonds under high temperature and pressure. A team led by Francis Bundy at General Electric back in the 1960s used large-volume presses to turn carbon into diamonds at only 15 kbars of pressure (although that reaction was very slow). Even higher pressures can be used to speed up the transformation, as well as certain catalysts (like nickel) to accelerate the kinetics. This is done regularly on a fairly large scale to produce synthetic diamonds for industrial applications. Note that Ben doesn't claim to be able to do this one "quickly" or "cheaply".