Does a "frequency-modulated" signal mean that the frequency isn't consistent?
A frequency modulated signal means that the frequency is shifted around a carrier frequency. This can be done digitally or via analog multiplication. 

What it looks like is a sine wave speeding up or slowing down. 


Source: http://www.justscience.in/articles/what-are-the-applications-of-frequency-modulation/2017/06/02

Yes, it normally has a specific bandwidth, some of the reasons being is radio licenses and bandwidth are in short supply (there are a limited number of frequency ranges). With frequeny modulation one can transmit more information in a given section of bandwidth. Most digital wireless communication uses frequency modulation.