Transfer power effectively from one coil to another through air
This is readily achievable and there is much practical material on=web. Search for inductive power transfer. 

Some good examples here but an annoying format. You can probably find the originals outside this system once you see something of interest here

An extremely worthwhile instructable. While he reports problems with operation he has images of inside equipment and circuit diagrams. An excellent resource

Many examples here

An instructable - rough but workable

Possibly same as above - somewhat different material

A hack-a-day example


  What kind of power can I hope to get through?


Watts is easy.
 10's of watts is not very hard.
 Kilowatts is doable 


  What is the optimal frequency? (is 50 Hz ok)


50 Hz is terrible.
 Higher frequency = smaller coils. 50 Hz ineeds vast coils or large cores.
 Best to use an assigned ISM (Industrial Scientific Medical) frequency and/or one used by one of the many systems that do this. 'Off the cuff' 125 kHz and 13.5? MHz are two such./ Searching on IPT as above will tell you more.


  Do I need to worry about radio polution?


Not if done well. The transfer is near-field magnetic - not RF.


  Will an unconnected iron core help?


Read the many articles. Coils need to be resonant. Use of an iron core is not usual.


  What is best - many or few windings?


Read the many articles - follow some examples cited above to start.


  Please also tell me, if you know of other reasons why this would not work.


Entirely doable. Start by copying others' examples.



ADDED:

Very good demo video.

Related website here

100% Duty Cycle: When Steven said "100% duty cycle" he meant "always-on AC waveform"