Overall, which case is the most convenient of the 3? Please explain your reasoning.
I've never used the Armor case, so I can't really speak to that one, but I will say that from what I've seen it's quite ugly and crazy huge. It may offer somewhat better shock protection than the lifeproof case, but it's too much of a trade off for me. I wouldn't want something that big (or ugly)
As for the two lifeproof cases, though, I have used the Fre since the day it was first available and I just picked up a Nuud case yesterday.
They both are fairly similar, of course, with the big difference of nothing covering the screen in the Nuud case. If you want a waterproof case, then you're going to need to make some compromises, but in my opinion, the Nuud case is far less of a compromise than the Fre.
The big concern with the lifeproof cases (and I would guess other waterproof cases) is the sound quality. Obviously the sound needs to travel through a waterproof membrane to keep the case waterproof and that membrane can introduce some distortion and muffling. Personally, I use a bluetooth headset 95% of the time when I'm on the phone and a bluetooth speaker when I want to listen to music. I rarely actually listen to anything right from the phone. If you listen to music on the phone a lot, or use the phone to talk to people, then this might be more of a concern to you. From what I can see, there have been some improvements in this area in the Nuud case. I played around with it a bit and to me it sounds a little better. I also had a few phone conversations without using my bluetooth headset and people didn't complain about my voice quality. It also seems like this can be kind of hit or miss depending on the individual case you get, so if you do have problems here it might be worth exchanging it for another one.
The other advantage that the lifeproof cases have over the other waterproof cases is that the mute switch is actually usable without opening a port in the phone... I believe most or all of the other water proof cases require you to open a port, then stick your finger in to adjust the mute switch. Not only does this mean that there's one more place that water could leak in, but it also means that you can't mute or unmute your phone when it's wet. Additionally, the lifeproof case comes with a waterproof headphone adapter, so you can use (waterproof) headphones while you're running in the rain or even swimming. Again, I believe most or all of the other waterproof cases require you to open a port to access the headphone jack and the case is not waterproof while that's open.
Finally, as far as the screen goes, in my opinion the screen protector on the Fre case was a pretty big compromise. They used some sort of an anti glare coating on the screen which actually affects the screen quality more than I would have guessed. I was quite surprised at how good the screen looked when I put the Nuud case on. Additionally, the screen protector doesn't always sit totally flat against the screen, so it can make it a bit of a pain to use the phone. This isn't to say that the screen looks terrible or is totally unusable with the Fre case... I used a Fre case for 8 months and was quite happy with it, but it still is a bit of a trade off to get a protected device.
As others have said, the screen is less protected in the Nuud case. If you spend a lot of time on the beach with sand or otherwise might get abrasive dust on your phone, then you might be better off with the Fre case. But, if that's not a concern to you, then I believe the Nuud case is the best waterproof case on the market right now.