RF Energy Harvesting Project
1) Which antenna is more suitable for this project? A half wavelength
  dipole antenna or a quarter wavelength whip antenna with a ground
  plane?


The choice of quarter wave or half wave depends on how good your ground system is.  If you can supply a good solid ground plane then the quarter wave will provide about the same level of performance.  A bigger gain would be with a directional antenna that you could point at the radiation source either actively by phasing it or physically by rotating it.


  2) What kind of end connection is needed? E.G. SMA?


Whatever fits your design, but SMA is good for high frequency use.


  3) How do I calculate the load impedance in order to calculate the
  inductor/capacitor value required for the impedance matching circuit?


If you are designing your own antenna you need a network analyzer to measure your antenna and fine tune the network.  Even if you aren't making your own antenna a network analyzer is the best way to verify your impedance match is correct.

Don't expect to "harvest" much power as you'll be lucky to get miliwatts.  Better off going for a large nearby FM or TV signal than WiFi