A vast array of potential applications is emerging for drones and other devices to collaborate from disaster relief and search and rescue missions to smart agriculture and IoT systems. As drones move across multiple altitudes, they must have the ability to communicate across any direction in a three-dimensional (3D) space. However, due to the heterogeneous nature of the drone body and its interaction with the mounted antennas, different antenna positions on the drone can result in variations in the radiation pattern. While there have been a fair number of airborne communication works, few consider the role that antenna positioning has on the resulting transmission along the azimuth and elevation planes. In this work, we study the effects of the drone body and various antenna placements on the radiation pattern and fading of drone-based channels. Through systematic anechoic chamber and in-field measurements, we show that the drone body alters the radiation pattern of the mounted antennas, rendering the common assumption of a constant azimuth radiation pattern invalid. In addition, the body increases polarization mixing of drone-based channels, resulting in significant degradation of the cross-polarization discrimination (XPD). Hence, we propose using effective radiation pattern and XPD values instead of relying on measurements and/or assumptions that disregard drone-antenna interaction. We then analyze the shadowing and losses associated with the drone body for many antenna setups at various elevation angles and show that when mounted on the opposite side from the ground transmitter, shadowing increases with relatively-higher drone elevations. To account for these body-induced effects, we introduce rotational loss that results in better prediction results of the large-scale fading behavior compared to conventional models that neglect these body effects. Then, we analyze the small-scale fading for various antenna setups and show that the Rician K-factor is strongly dependent on elevation for polarization-matched vertical links, while it is approximately flat for cross-polarized links. To do so, we conduct a set of drone-to-drone (DtD) experiments at high altitudes with no surrounding reflectors and compare results to those obtained by our ground-to-drone (GtD) measurements. We find that, while at low elevations the ground can reduce the K-factor by 10 dB, at higher elevations, small-scale fading is dominated by the antennas, not the ground.