You are using the term "DC" in the wrong context here. The transmitter is not transmitting DC, it is simply transmitting with no modulation. On a similar note, many people use the term AM band. It is often used to describe the Medium wave band, but can cause confusion, as amplitude modulation is used on Long wave (LW), Medium wave (MW), & Short wave (SW). So I never use the term AM to describe a waveband. AM is also used on the Aircraft VHF band (118 - 137MHz).
A lot of receivers that are able to pick up morse code (CW) are called Communication receivers. The beat frequency oscillator (BFO) is a circuit within the receiver that, as the name implies, generates a signal that mixes with the incoming signal to create a beat frequency that's in the audio range, so can be heard.
You can hear this effect yourself if you tune your receiver to a radio station (if there are any still broadcasting in your area) & then gradually tune the transmitter closer to the radio station. You will hear a high pitched tone, gradually getting lower in frequency as you get closer. When your transmitter is on the same frequency as the radio station, you will probably hear a fluttering sound because a simple transmitter like that won't be able to tune exactly to that of the radio station because it doesn't have very good stability. When a transmitter is on exactly the same frequency as the radio station, it is known as "zero beat", as the frequency difference between them is zero, so produses no difference tone.