How do lasers work if the phase shift from reflection is 180 degrees? I am having a hard time wrapping my head around the phase shift that is imposed by reflection. I'm specifically thinking about things like visible light of off mirrors or RF reflecting off of silver, gold, or copper.
How is it that a resonant cavity, laser, or an antenna be tuned to increase constructive interference (and the converse by decreasing destructive interference) when many of the properties of these types of things rely on reflections? This is particularly apparent with lasers which use mirrors on either end.
I'm almost certainly confusing many different phenomenon right now.
As a thought experiment, if I have a hollow tube that is reflective to RF that has a reflective cap on one end and an RF source on the other, and I put a strictly receiving antenna in the center (or as far as I can tell, anywhere), will a usable signal be detected? I'm assuming that the antenna doesn't affect waves that pass by on the way to the capped end and that the walls are lossless. Let's also assume that the RF signal is of a single wavelength.
Based on my understanding of reflection, the phase would be inverted 180 degrees, which should result in complete deconstructive interference. Thus, there should be no detected signal. Is this correct?
 A: Not only that but that technique you described is the idea behind the so-called slot line. You take a homogeneous waveguide, be it coaxial or simple, rectangular, circular, and you cut  a narrow slot along the guide into which you put a short monopole whose "ground" is the skin of the waveguide and connected to a detector diode. As you move the probe (antenna) along the slot you can map the electric field intensity of the standing wave that is the result of the sum of the traveling wave from the source and the one reflected from the other end. As you move the probe along the slot you can measure the standing wave ratio it being the ratio between the maximum and minimum voltages measured along the line, see 1. This electric monopole, if both the probe and slot are sufficiently narrow responds "only" to the electric field.
If both the line and the termination are lossless reflector then the maximima/minima will be separated exactly by $\lambda_g/4$ and their positions depend only on the terminating impedance. If your termination flips the phase of the E-field then at that point the probe will measure zero and also every $\lambda_g/2$ from that short, half way between them will be the voltage maxima.
Nobody does this in practice but you can measure, if you wished to do so, the magnetic field with a small loop put through and moved along a properly placed slot where the magnetic field has maxima.
A: 
if I have a hollow tube that is reflective to RF that has a reflective cap on one end and an RF source on the other, and I put a strictly receiving antenna in the center (or as far as I can tell, anywhere), will a usable signal be detected?

Yes, unless you get unlucky about the placement of the antenna.
The forward and backward propagating signals will form a standing wave. Since you chose a low-impedance reflector (producing 180 degrees phase shift), there will be a node (null) in the standing wave pattern at the cap, then an anti-node (field maximum) 1/4 wavelength in front of the cap, another node 1/2 wavelength in front of the cap and so on. Right back to the generator. And if there happens to be a node right at the generator you have a good chance of blowing up the generator since it will think it's trying to drive a short circuit directly.

Based on my understanding of reflection, the phase would be inverted 180 degrees, which should result in complete deconstructive interference. Thus, there should be no detected signal.

Right at the cap, this is correct. There is destructive interference and a zero in the field strength.
But slightly away from the cap, the reflected signal will be phase shifted from that initial phase at the cap, as will the incident signal, and the interference will not be fully destructive (and may even be fully constructive) unless the distance is a multiple of 1/2 wavelength.

How is it that a resonant cavity, laser, or an antenna be tuned to increase constructive interference ... when many of the properties of these types of things rely on reflections?

Remember that visible lasers operate at wavelengths measured in 100's of nanometers, but the cavity length is typically measured in centimeters or even meters, so there are many, many wavelengths inside the cavity. That means if one wavelength (say 632.8 nm) is not perfectly resonant in the cavity, another very nearby wavelength (say 632.805 nm) will be resonant. But the gain medium and other contributors to the laser operation (in a simple design) aren't so selective that they will work at 632.8 but not at 632.805 nm, and you just end up with a laser operating at a slightly different wavelength than you expected (unless you tune your laser very carefully).
