Design considerations for bifilar coils for Wireless Power Transfer
What are the design recommendations for the Receiving coil (given above constraints)?
  


Designing the coils for wireless power is a rather complicated topic. You can find heaps of research papers on the topic. It depends on many parameters such as your working frequency, sizes of the coils, transfer distance, power level, target efficiency, etc. If you are after the high-efficiency, then you should maximize coil quality factors (\$Q_{\rm{TX}}\$ and \$Q_{\rm{RX}}\$) and the coupling coefficient (\$k\$). The figure of merit for high efficiency is \$k^2 Q_{\rm{TX}} Q_{\rm{RX}}\$.  You should minimize your coil resistance to maximize the coil quality factor and increase the coupling between them, which will again be a complex optimization problem.  There are some options for you to choose: The use of Litz wire to reduce skin effect losses, and optimize the number of turns and the gap between tuns to optimize \$kQ\$ product. For instance, you can find such numerical optimization in this paper.


  
  What are the effects of the two winding approaches, and which one is more suitable for my application?
  


For the winding of two turns, one strand on top of the other is better because you can use the available space effectively. you can have more turns for a given outer diameter of the coil to have higher inductance (and mutual inductance), and/or you can introduce a small gap between turns to reduce proximity effect resistance. 


  
  How the coil interconnection approaches affect the overall power transmission efficiency and which one is more suitable for my application?
  


The logic is pretty simple. You should make sure current in the winding flows in the same direction so the effective field from both windings added up. Otherwise, the field created by two windings will be canceled each other.  

If you use the left one, you should connect your source between terminals 'x' and 'y' so that two windings are in series - the effective total number of turns will be 2N.

If you use the right one, then you should connect the source between terminals 'y' and 'x-z connected together', so that two windings are in parallel-the effective coil resistance will be half.


  
  Should I have thicker wire with less turns or a thinner wire with more turns for the transmitter coil?
  


This is again a complex optimization problem for maximum \$Q=\omega L /R\$. A higher number of turns will increase the coil inductance \$L\$ (having a positive effect on \$Q\$), but it will also increase coil resistance \$R\$ (having a negative effect on \$Q\$). If your working frequency is low (say, below \$50~\rm{kHz}\$), then you can assume the proximity effect resistance is negligible. Then you may approximate \$L\propto N^2\$ and \$R\propto N\$, which makes a higher number of turns a better choice. 


  
  In both connection scenarios, I end up with three points (x,y,z) to which I could connect the driver circuit. What are the connection approaches and driving options that are valid?
  


refer response to Q3


  I found several examples that use a single transistor and a battery to drive the transmitter coil, but I was hoping for a slightly more complex and effective suggestion.


Now you are referring to the configuration of the driving circuit where you have many options. I suggest the use of a current fed push-pull oscillator, for example, the circuit configuration given in this application note can be a good choice. 


  
  Is the spiral receiving coil (shown above) compatible with a bifilar type transmitting coil?
  


Yes, provided that you configure bifilar coil correctly.