There's your basic answer--it was right in your own message!
1. Oxidation will attack beer over time--it's especially pernicious with regard to hoppy beers, which is why often the advice is to drink them fresh. If your beer isn't lasting long, you may not have had it around long enough to notice.
2. Ever been at a keg party where they used a picnic tap and a hand pump to dispense the beer and then had some of that beer the next day? Oxidized! It may be the best example I can think of to find oxidized beer. Too much O2 post-fermentation = ick.
3. I think there's value in doing this: every time you brew, try to do something better. Find ways to limit O2 exposure recognizing that without a specific system you probably can't eliminate all of it. But you can eliminate most of it. Just keep adopting process elements that do that.
How?
A. Purge keg w/ CO2. You can actually hook up a line from the airlock to a QD attached to your keg, release the PRV, and as the beer ferments the CO2 it offgases will be fed into the keg. If the gravity is high enough you'll produce enough CO2 to purge the keg. Every 2 points of gravity is equivalent to about 1 volume of CO2. So if your beer starts at 1.060 and finishes at 1.010, you'll have produced about 25 volumes of CO2. Purge a keg 25 times with CO2 and what do you have left? CO2.
You can also purge it with bottled CO2.
B. Just because I'm fairly anal when it comes to this stuff, I'll typically do a Star-San purge of my kegs after cleaning them. That means a full keg of Star-San has it pushed out by bottled CO2 into another keg using a jumper from OUT post to OUT post. That leaves almost all CO2 from the bottle in the keg. Then I'll harvest fermentation CO2 from the fermenter to push into the keg and I have an almost perfectly CO2 keg.
C: Rack into that now-purged keg through a QD connected to the OUT post. You can do this with gravity--I've done it that way many times. I will then feed the displaced CO2 from the keg--coming out through a QD on the IN post--back into the fermenter so instead of drawing air into the fermenter as the beer drains/racks into the keg, I'm drawing CO2 from the keg back in.
D. There are many ways to avoid suck-back of air/O2 into the fermenter at cold crash. One is to do the balloon trick. I wasn't sure I could get a good enough seal so I used a breadbag filled with CO2 and twist-tied around the airlock. If you ferment under pressure, there's already CO2 in there to account for the reduction in headspace at cold crashing. BobbyM from brewhardware.com sells something that does this, it's fairly cool IMO:
https://www.brewhardware.com/product_p/ccguardianv2.htm
I have some pics below showing how I've done this. I now have a Spike CF10 conical I'm crashing in and fermenting the last bit under pressure, so there's no suck-back. I show how I'm feeding CO2 off the fermenter into the keg.
There are lots of ways to do this. Just keep moving toward doing it better. I still am.
The Breadbag method:
View attachment 585548
O2-free racking feeding CO2 back into fermenter. I just used an airlock w/ the top cut off, it fit the tubing I had. I now use a piece of rigid plastic tubing from a bottle filler through a drilled stopper.
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I now have a Spike conical. This pic shows me feeding CO2 off the fermenter into the keg into which I'll rack the beer. If you look closely, you'll see I've hooked up a jumper in series so that the O2 is purged from that too, so when it comes time to rack into the keg, the jumper I'll use is free of O2 as well.
View attachment 585552
The device I use to connect two liquid OUT Quickdisconnects.
BobbyM sells it at Brewhardware.
View attachment 585551
The jumper:
View attachment 585550
I've done a TON of brews without one and always use a blow-off and I've never (at least to my observation) have seen the liquid getting sucked back in (although it's gotten close) and the beers have been great, but always looking to improve on my methods especially in guarding against O2 post fermentation. 
