A brownout is when you have a low voltage condition. Your lights will dim but they won’t go out. Fans might not have enough torque to keep spinning, which can be a big issue in electronics where the electronic device stays on (maybe just sorta) but it’s cooling fan stops. I’ve lost more equipment to brownouts over the years than to blackouts.
Blackouts are when the power goes completely dead.
You are indeed experiencing what is called a transient fault. If you overload a circuit in your home, the breaker trips and the line goes dead. Then you trudge down to the basement and figure out which breaker tripped, and turn it back on. The power company has the same thing, except that hiking a few dozen miles to figure out what tipped is a bit more of a pain than going down your basement stairs. So their protective devices have what are called “automatic reclosers” on them. When there is a fault, the recloser turns the line back on. Reclosers are programmable. Usually they are programmed to turn back on a couple of times fairly quickly (within a few seconds) and then they’ll wait a bit (maybe a couple of minutes) and then they’ll try one last time before giving up. If you have something like a lightning strike or a tree brushing up against a line, the fault usually clears fairly quickly, the recloser turns the circuit back on, and all is well. If the recloser faults for its final time, it’s done, and the power company guys get to drive out and figure out exactly where the fault it.
So basically it sounds like you are getting some sort of transient fault and then the recloser is turning the power back on.
One thing about deserts is that they are dry, which may seem like a really smart Captain Obvious thing to say, but where I’m going with this is that the electrical grounds tend to suck. Really dry soil isn’t very conductive. This means that protective grounds, like those that help shunt lightning strikes harmlessly into the earth, don’t work as well. Does lightning accompany these rain storms? Lightning can travel a long way down power lines, especially if you don’t have good earth grounds to shunt it harmlessly into the soil.
Another thing about deserts is that, with the exception of places like Las Vegas, not many people live there. The sparse population density means that power lines tend to stretch longer distances, so a fault a good distance away from you can still take out your power.
I asume you mean UPS (Universal Price Codes don’t help your computer much 
). When the battery starts to die, the UPS just won’t provide power for as long. If the battery goes really dead, the UPS will often detect it and will beep, if it has a battery monitoring circuit in it. If it has lead acid batteries, they tend to chemically self destruct if you discharge them too much, so use the UPS to ride through short faults and use it to safely shut down your computer for longer outages. Don’t run it until the batteries go completely flat. Often they’ll have an alarm that will let you know they are getting low on battery power. The less expensive units don’t have to much monitoring or alarming.
If your UPS uses lithium ion batteries, those just die an early death no matter how well you treat them. They are small, lightweight, and have an excellent energy storage capacity, but they don’t age well. If they run hot they die an even earlier death. At best, they’ll last maybe 5 years. If they run hot, they may only last a year or two.
Ni-Cad batteries are heavier and have less energy density than lithium ion, but they don’t die as quickly (especially if you treat them well) and they don’t self-destruct if you run them dead.
Less expensive UPS systems don’t have much in the way of noise and spike filtering. Sometimes the spike filtering is just a few metal-oxide varistors (MOVs). Those work by creating a dead short to ground when the votlage gets to high, which clamps a lightning strike to ground and prevents it from doing damage. The MOVs get destroyed in the process though, and cheap UPS systems (as well as power strips, which also use MOVs for protection) give no indication that they have failed and that you have no more protection. The better ones will have an indicator light that let you know that the MOVs are still intact.
Note that surge protection has its limits. A cheap power strip might be rated for a few hundred joules. A better surge protector or a UPS might be rated for a few thousand joules. A high end UPS might be good for a few tens of thousands of joules. A lightning strike contains a few billion joules. There isn’t much that you can buy (at least not affordably) that will protect you from a direct lightning strike. Those are fairly rare, though. More often the lightning will hit a power line somewhere, and you’ll get only a fraction of that energy, which a good surge protector will be able to protect you from.
When looking at surge protection, the lower the clamping voltage the better, the higher the joule rating the better, and the higher the shunt current rating the better.