Does a unit like this detect voltage fluctuations in a room? I am experiencing some light bulbs flickering in the same room.

The Klein Tools Cat. No, RT250 GFCI Receptacle Tester with LCD refreshes the voltage reading 3 times per second, but you really wouldn’t be able to see the fluctuations in most scenarios. That would require something like a meter with oscilloscope feature to measure more times per second.