Need of Baluns in Radio Frequency Integrated Circuit

If I remember right, narrow band FM is used. "Narrow" refers to the small maximum frequency deviation from the carrier center frequency. In FM, more deviation means more signal to noise ratio. Less fidelity is required to send simple digitally encoded signals and voice weather reports than good quality music as commercial FM does. In this case, the deviation is 5 kHz (again, if I remember right).

As for how to build a receiver, that's no different than any other FM receiver. There are various topologies for that. At 160 MHz or so, you have to wake up and design such things carefully. It's not like 1 MHz AM where simple hacks can get reasonable results.

I don't think not using a balun and just connecting one side to ground is an option. As your signal is defined differentially, V = (V+)-(V-) having a DC component at one side and 0V at the other side will utterly destroy your intended calculation. You do need a single ended to differential converter, and mainly the two methods of achieving that are using precision amplifiers for low frequency applications, and baluns for high frequency applications. Low hundreds of megahertz is quite a high frequency, for which you'll almost certainly need a balun to operate this multiplier.

Also, I'd strongly advise you to consult an experienced EE for this design. Designing anything in the hundreds MHz range requires experience in board layout and signaling, and while recent standards operating at the GHz range may make a 100~200MHz design seem trivial, it certainly is not.