The Micromitter has flea power. Only -10dBm out of the chip. To get 1W you will need to amplify at least 40dB and deal with the resulting harmonics! :)
RF feedback is known to cause "hum". This is normally not a problem when the transmitter is mounted in a metal box and screened audio cables are used. Kits that comes with plastic boxes may have this problem.
I don't know about a single chip apart from a micro. Discrete it could be done with a row of D-type flip-flops wired as serial-in-serial-out (SISO) shift register and clocked by the 555. You may also use 2X 8bit shift register chips with the proper feedback after 10 stages.
Extra leg may also be for mechanical strength when mounted in a pcb and not connected to anything internally. Few simple measurements will be all that is needed to solve the mystery.
You used to get split capacitors (2 caps with one common neg terminal) used in older TV applications, although I have not seen them for quite some time.
Here in the US it is quite common. You buy these inserts and install them into a normal light switch box. It has 3 positions; off-auto-on. The circuit has a sensor to sense movement inside the room and the whole thing costs about $6 and takes about 3 minutes to install.
There's LC filters of all kinds, low/high frequency and of various power levels. For example, you won't use toroid at very high frequencies because of the losses. Each applicarion will require its own selection.
Souns like a easy task for a PIC micro. Look on the microchip website for typical counting applications and code to drive the displays. You will also find a lot on Google.
Often these TV's have a switching type PSU. They use a high value start-up resistor to get the PSU control circuit going. After that DC gets fed back through a diode blocking circuit to supply the control part of the PSU. Over time these resistors increase in value and have problems supplying...
That you can calculate from the well known formula wl=speed of wave/frequency
wl = full wavelength in m
speed = 3 X 10 ^8 m/s
frequency of operation in Hz