Well, 50 Ohms is an aproximate value. If I recall correctly, an open dipole in free space has arround 72 Ohms in the center. But under some practical circumstances, 50 Ohms is quite real.

This antenna has the better efficience (considering the distribution in space). It means that you can design a directional antenna with a better efficiency("gain") in some specific direction. The "gain" in dB refers to an isotropic vertical dipole.

Joe, if I understand correctly you questions, a transmitting antenna, for example, could eventually transfer 100% of input power to space, even being shorter than Lambda/4, providing adequate adaptation. This is theorical, because any parasit added element is not perfect (finite Q). But even then, antenna will be inefficient, because there is a missmatch between its radiant resistance and free space "radiating resistance". I am not sure really if it is the way to put this fact...

EDIT: Correction. In fact, the problem is that very short antennas have very little irradiation resistance. In this case, any practical matching element with finite Q will introduce losses, even worst as shorter the antenna is. Take a look at http://ecjones.org/hightc.html, for example.
END EDIT

So, matching a short antenna will improve efficiency basicaly by reducing SWRR ( standing wave reflecting ratio), or matching impedances (different words but same concept), but still efficiency will be less than a theorical dipole.

And receiving signal deppends of the field strength produced by Tx antenna, and Rx antenna efficiency. Tx antenna impedance has nothing to do.

Daniel

PD: you can design you receiver input impedance to be iqual to the conjugate of the antenna impedance (if the antenna Z=10-j200 Ohms, you make your receiver input 10+j200 Ohms). But this IS "matching"...