I will try:
Most mixer circuits have two inputs, usually the signal you want to monitor and the local oscillator. There are several types of mixer, some using cancelling techniques (balanced and double balanced mixers) to eliminate some of their output signals but for simplicity, the ones normally used in domestic receivers are not balanced. They produce typically four output frequencies:
1. a small amount of input signal leakage
2. a small amount of local oscillator leakage
3. a signal at the sum of the signal and local oscillator frequency
4. a signal at the difference of signal and local oscillator frequency.
You can use 3 or 4 as the signal you want to select, the choice is yours, the only difference it makes is whether the LO is on the high side or low side of the frequency you are receiving.
You can look at 3 and 4 the other way around - for a given local oscillator frequency, the output frequency (the IF) can be made from two different input frequencies, it is up to the selective circuits at the input of the receiver to block the unwanted one as much as possible. You will come across the term "image rejection" used to describe how well the unwanted input is reduced relative to the wanted one.
Regarding the frequencies that can carry information, yes there is a relationship but one isn't directly proportional to the other. Modulation is the name for the process of adding information to a carrier, in doing so it produces a shift in the phase (PSK, QPSK etc) the frequency (FM etc) or the amplitude (AM, DSB,VSB, SSB etc). All of these make the required bandwidth greater than a single unmodulated carrier. The amount the bandwidth increases by depends on the kind of modulation and the amount of information the modulation carries. It follows that you can't use a low frequency carrier to carry a wide bandwidth signal, it simply runs out of space in the radio wave spectrum. In general, as long as the carrier frequency is high enough to carry the modulation it will work but given there are so many users of the spectrum it makes sense to keep the low frequency bands for narrower signals and higher frequency bands for wider signals.
For example, a digital TV transmission channel is typically 27MHz wide, it wouldn't make sense to place it in the 0.5 - 1.6MHz AM radio band! There are still many narrow band signals on higher frequencies though.
Brian.