A mixer is just a circuit that multiplies two signals.
The purpose of a mixer is to shift a signal in the frequency domain.
Normally a mixer multiplies a signal carrying information (some range of frequencies, e.q. received signal or signal that is going to be tranmsitted) with a signal without information, consisting of exact one frequency (from the local oscillator).
An I/Q-Mixer consists of two ordinary mixers that each multiply the signal carring the information with the local oscillator signal, but one being 90° phase shifted.
A demodulator in general is a circuit that extracts the actual information of an RF signal.
There are many different demodulators (as there are many ways of modulation) and even for one particular modulation (e.g. FM, SSB) sometimes more than one type of demodulator is possible.
An I/Q demodulator is just some kind of demodulator using (among other subcircuits) an I/Q mixer as mentioned above.
Summary:
An I/Q mixer is a part of an I/Q demodulator.
An I/Q demodulator always contains an I/Q mixer, but I/Q mixers may also be used in circuits that are no demodulators (e.g. modulators).
I'm sure there are many methods to recover frequency and phase of a carrier, but one that is widely used in analog TV, and also in some digital communications, is the one involving a burst PLL.
The signal transmitted can be divided, in the time domain, in units. Each unit contains several regions (also separated in time), which will depend on the application. For instance, in analog TV, each unit is a horizontal line, and contains 3 regions: sync, color burst and luma+chroma information. The sync region is used to distinguish where the units begin and end. The luma+chroma information is the actual video infomation you want to transmit. And the color burst is a "windowed view" at the carrier that was used at the transmitter, to encode the luma+chroma information. During the color burst region, the signal transmitted reflects only the carrier. A multiplexer directs, during each region, the corresponding signal to the output of the transmitter.
The piecewise signal transmitted, for the case of analog NTSC TV, would be this one.
At the receiver, a module finds and separates the sync information embedded in the incoming signal, and uses sync-derived timing to close a gate that directs region 2 of the incoming signal (i.e., the color burst) to a burst PLL. A burst PLL is a heavily damped PLL (it has a loop filter with a long time constant -much longer than the duration of one "unit") that is allowed to modify the voltage that governs its VCO only during the time window while the burst gate is closed. As a result, the output from the burst PLL is a sinusoidal signal that is locked in frequency and phase to the incoming color burst, and that exists always. Even during regions "sync" and (this is the wanted one) "luma+chroma". Thanks to that recovered carrier existing always, it can be used to synchronously demodulate the incoming luma+chroma information.
I've talked about analog TV, color burst, and luma+chroma, but the concept behind a burst PLL is much more general, and it is in fact used in some digital comunications.
The following figure shows what could be a general block diagram, not specific to TV, and which deals with the general signals I and Q.
Best Answer
So I've done some research since I have a similar problem and everything leads me to fully integrated quadrature demodulators like LT5517 with a good NCO if you need AFC. All digital systems with direct sampling might have even better noise performance especially if you use oversampling (source) but they have a detection delay, so your application should tolerate this delay if you want to use this method. Search for FPGA or DSP FM demodulators, there are plenty of articles. The best solution for data transmission I've found so far are specialized transceivers with interference resistance like AD9364, but those come in 144-LFBGA and cost $210 per chip. See this article for more info.
If you need a simple and decent demodulator, conventional quadrature demodulator has the best noise performance if you use high quality parts for it.