Yes. There are a few common ways this is done. Rectennas are used for high frequency waves and magnetic coils (think wireless power chargers) are used at low frequencies. Unless you are intentionally beaming power at a rectenna, there will not be a lot of power available, but enough to run some simple circuits, especially if you duty cycle your circuit (turn it on and off).
Yes, this is possible. Best to have a resonant receiver (applicable only for narrow-band signals). As already stated, magnetic coils are best for relatively low frequencies (see eg. Qi receiver circuits).
Power available is normally really low (μW and below). And you should be aware that this energy comes from the field - weakening it. Thus, when 'vamping' a WiFi signal, WiFi reception 'behind' the receiver will be weaker than without such a receiver.
Low frequency systems usually do not use radiated power - they are in the near field - and use magnetic coupling. They are transformers, not antennas, if by antenna you mean something that picks up a radiated field.
A transformer system can be much more efficient than a radiating system because a radiating system loses all its power into the distance if it is not used, whereas a transformer system doesn't lose (much) power to its surroundings and only loses power to internal losses and the, for example, tuned RFID tags that are designed to interact with it.
RFID uses the principle and is very effective.
A long time ago, I heard that someone built an antenna array to pick up significant power from radio or television broadcast energy and use it as a power supply, and that this is illegal (it leaves a shadow behind - though not for a huge distance, because of diffraction).
The problem with harvesting of radiated power is that it is ok for one or two devices, but eventually with mass use the harvested power reduces the performance of the system it is harvested from, so that, for instance the broadcast range or coverage is reduced.
Many have answered your question, the best example is our radio &TV system.The transmitting antenna as powered up with 1-10 KW of power.Whille receiving we get signal strength in terms of mili or micro watts.That's why such signals are amplified before processing. Though we can get some amount of energy,the system efficiency is very very low.
An RFID system using e.g. two large coils at the exit of a shop is not a radiating system at all. It only radiates a small amount that is totally irrelevant to its operation and would be eliminated if possible. It only couples to the receiving devices by transformer operation, using the near-fields whose energy density falls off as the inverse fifth power, and so cannot carry power away (each successive radial shell has less total energy in it).
This inductive operation is not at all the same as the interaction between two antennas by the radiated field that may interact because the re-radiated field from one reaches the other, or between a shell of antennas around an antenna that may re-radiate back to the central antenna and change its impedance.
For the travelling radiated electromagnetic waves, the energy of electric field is equal to the one of magnetic field. Both fields must coexist when the electromagnetic waves propagate in the space. Only electric field or magnetic field can not be radidated in the wave form. Therefore, it is impossible to harvest the electric or magnetic field energy independantly.