How do I power a device with a 4-directional 4-pin connector? I have an idea and I would like you to point out what's wrong with it
1) Are there easier ways to do this? Are there readily available
  sockets or designs for my purpose?


Connectors can be hard, usually searching a distributor like digikey in the connectors section is the best way to find the right connector. Another thing that might be valuable is to find a distributor catalog and look through it to get an idea of what's available. 

Even a 2x2 0.1" header might be best for your purposes, or 4 seperate 0.1" headers. The real question is: do you want the connector to be the mechanical link, or a fastener?


  3) How to choose the right R and C values for a given frequency?


RC filters form either low pass or high pass filters. High pass filters block DC, so only AC currents can pass through them. Filters block frequencies.

Either filter's cutoff frequency is at 

$$cuttoff = \frac{1}{2\pi R C}$$

If I wanted to keep 10Hz and block everything over 100Hz then I'd make a low pass filter with a cutoff somewhere in between. Then I need to choose RC values, Lets say I have a 1k resistor, so I start there and solve for C, I'll try 20Hz. I get a C value of 0.8uF but those capcitors are hard to come by in a value like that so I round it to 1uF. Then I solve the cutoff for an R of 1k and a C of 1uf which would give me 15.9Hz then I decide if that would be acceptable for my requirements.