What is CP for pokemon?

HI Muxaul!
I think a possible answer is that CP is a general indicator of how good one pokemon is but it is not saying that a higher CP pokemon can absolutely beat the other one in all cases. Like what you said in your example things like type and gym mechanics are also important.
If two Hypnos have the same CP, it's fairly to say that they are ''equally strong'' but the way you use them in fights also matter. Different IVs lead to three different things in my mind for this case. Those two Hypnos may have their own specialty like one is better at attacking than the other. And the stardust you need to power up them to the same cp could be different. And their potential CP in the future for even more power ups is also different.
For the last question, choosing a defending pokemon should not be only on the cp. Analyse of that can be seen in the gym defender tier section. I just found this website couple of days ago, contents here are extremely helpul. You should definitely look at that section. Good luck and catch them all!

It took me awhile to understand CP, but what I've come to realize is that IVs are almost irrelevant. Not to say that they have no impact on the game, because they do, but how much I feel like is a bit exaggerated. Let's say I have two squirtles, one with 80% IVs and another with 90%. Now let's say someone had the stardust and candy go in hypothetically power up and evolve both of them. Even though the 90% squirtle is better, how much when it's evolved and maxed out is low to negligible. You probably wouldn't even notice a difference aside from moveset which is a totally separate entity in itself (since you only ask about CP, I'll go no further). Of course you may notice if it's way lower in IVs, but again, low seems almost subjective to the Pokémon you're working with. However, let's go back and say that the 80% squirtle has more CP than the 90% one, which means it's at a higher level. If the difference in level is great enough, most people would tell you to just stick with the 80% one. Yeah, it's not as good, but in the long run you'll spend less stardust, candy, and most importantly time. CP is more of a reflection of base stats which is why only a few Pokémon remain overused; their base stats greatly overshadow the others. The problem with Pokémon like dragonite is the fact that Pokémon that counter it face more problems than they should, simply because their base stats (and thus their cp) fall way under it. This is a problem that I think will fix as more generations and Pokémon come.

To answer your gym question a bit more directly, remember the above and add into account AI. Players put higher cp Pokémon into gyms so that however few hits they get in count.

1. Yes, higher CP pokemon have higher attack/defense/stamina stats, and are thus "stronger" than lower CP pokemon. CP is a value equal to a pokemon's stats x its level (the full formula can be found somewhere around here): having higher IVs means a pokemon will have a higher CP, and having a higher level also means a pokemon will have a higher CP. Thus, to get the maximum CP possible, you ideally want a pokemon that's 1. A high level, and 2. Has high IV values.
2. CP isn't everything. Tangela can take out higher CP Vaporeons because of its elemental typing (grass is super effective against water). Also, with proper dodging and timing of attacks, a 1000 CP Vaporeon could maybe take out a 2000 CP Vaporeon - it would just require a lot of talent.
3. Two Hypnos with the same CP will be more or less equal in battle, assuming they have the same quick and charged attacks. IVs only matter if you want to level a pokemon up to level 30+. Think of it this way: having higher IVs means a pokemon will have a higher stat cap, which means a higher maximum potential CP value. A level 25, 0% IV Hypno might have the exact same CP as a level 22, 100% Hypno, and will perform the same. However, if you level both of them up to level 30, your 100% IV Hypno will now be better because it will have a higher CP.

CP is only a first guess. Nothing precise. Especially if you compare different pokemon with different movesets.
And as you noticed, the attacking pokemon can usually be quite weaker CP-wards and still win. (See your Tangela-Vaporeon example; the lower-CP Slowbro is a better defender than Vaporeon, but this depends on movesets as well; and your Tangela has a big type advantage against Vaporeon).

For comparing e.g. 2 Vaporeons with identical moveset CP is quite ok. (you can push this to pokemon with almost equal base-stat, identical moveset and equal typing) But even then it is flawed.
Reason: For the real performance all 3 real stats (after IV) should be weighted allmost equally (sta/HP slightly higher, def slightly lower). But in the CP-formula sta and def are only weighted with their square root.
Thus Dragonite has highest CP, but is still weaker than Snorlax (no matter if for attack or defense, time+potion usage ignored).

Example
When comparing the 2 Pidgeots... (Pidgeot, because of almost equal base stats for att/def/sta: 170/166/166) Pigeot-A might have IVs att+15,def+0,sta+0 and Pigeot-B might have IVs att+0,def+15,sta+15.
They would have almost equal CP (B slightly - 0,2% - higher) if at same level. But B would be 9% stronger than A.

Comparing different pokemenon:
For attack we also have to consider time and potion usage. Dragonite outperforms Snorlax here by far and is thus the better option for attacking gyms. For 1-on-1-fights Snorlax might be the better option again.

And for defense you also have to consider the metagame. High CP defenders discourage attackers. A CP3000 Dragonite would, a CP2000 Slowbro not. The Slowbro might still be the harder defender (depending on moveset).
And your good defender (Slowbro) might be kicked out quickly where the bad at defense Arcanine stays in the gym.