How do I figure out the ohms of an antenna?
A whip antenna, in theory, is resistive at 1/4 wavelength (perhaps slightly shorter in practice) when it is mounted on a ground plane. At 100 MHz the wavelength is about 3 m so a 1/4 wave whip is going to be around 75 cm. So this antenna at 45 cm is too short. You will also need a ground plane or radials to approximate a ground plane. At resonance the theoretical resistance is about 37 ohms, which is usually considered close enough to 50 ohms. As @mkeith says, the match will worsen away from resonance, being capacitive below and inductive above. Probably need to retune for different frequencies. 
You don't need a balun for a whip. Just connect the outer of the coax to the ground plane at the base of the antenna. On the other hand, a dipole is balanced so you need a balun for best performance. A half wave dipole is about 75 ohms at resonance, also probably close enough to 50 ohms in practice. A balun, as the name suggests, is usually a balanced to unbalanced transformer on a small ferrite core. You can also wind a few turns of coax onto a ring core for a reasonably good balun. There are plenty of examples out there.