Are IV really important?

Well, technically IVs are kind of important because the higher IV, the higher CP (not that much, tho) the mon will have.

The higher CP the mon have (compared to the same species) the most chances of sitting higher in a gym you have.

Basically, the real 'impact' of the IVs in the game is pretty much this, being on the top a gym despite having the same mon in the same gym.

For attacking mons, the IVs are basically useless.

Now, that was the technical explanation.
You also have the "fun" aspect. It if weren't for the IVs this game would have turned completely boring 3 monthes ago. All this IV thing adds the game a 'fun and challenging' side, that will eventually makes you wanting to get higher and higher IVs mons you like.
If you are not the kind of people that is being attracted by this or drove by this in order to get some extra fun, then I don't know why you are still playing imo.

It's exactly the opposite. High IV is theoretically worth less on a Pokémon that already has high stats, compared to Pokémon that has lower base stats.

For example:

Dragonite's base stats are 250/212/182
15 stat points for each of the stats is proportionally a lot less than for example Cloyster that has base stats of 196/196/100.

The difference with 0% IV and 100% IV is not that big, movesets are a lot more important. However, higher IV still means higher CP which means that the Pokémon will sit on higher spot in a gym making it harder to remove from there and making it more likely to stay there for longer period of time.

EDIT: Looks like Emiliano55 was faster this time :) But the 'fun aspect' is a good addition too. Myself being pretty OCD, I enjoy a lot about this technicality in the game.

The way I look at it is to go to level 40 of the pokemon and compare the min and max CP. That distance can be around 400 for some strong pokemon.

So consider your IV values. Every 25% of IV would then be about 100CP at level 40 (like 75 at level 25 let's say). Basically, 100CP is actually a pretty big difference. It can be 2-3 spots higher up in a gym. It'll be more intimidating, etc...

So while the difference between 75% IV and 100% IV really doesn't matter all that much (75CP), where do you draw the line? 150CP? 225CP? That's a personal decision, but to me if I have the choice I'll spend some dust and get the extra 100CP if I can, so anything under 70% IV I think long and hard before powering. Movesets play a much more important role (especially when attacking, as defending you may just not get attacked).

Highly depends on your level, the attackers/defenders you have and rarity/candy supply.

Of course I spended a few power-ups to get my not so good dragonite above 2500CP. But in general I saved my stardust.

Now I have 3 dragonites and many other good options for attack and defense (well good is relative for defense). So I don't invest in low IV or bad moveset any more. And low IV is different for lapras and hypno...

But you should really consider IV when powering beyond lvl 30.
Dragonites with att+7 and +15 (other IV identical) differ by about 3% in performance and CP, but above lvl 30 this are 4 power-ups CP-wise. Corresponds to 20-60 candy (increasing between lvl 30 and 40) and 22-40k dust.
For Dragonite and sta (again +7/+15) the difference is CP-wise lower (2%), regarding performance it is 4%. Again not so important, but expensive to compensate with power-ups.

Nonetheless bad movesets reduce performance by max. guessed 20-35%, bad IV (rule out the sub 50% pokemon, consider only final evolutions, ignore chansey) reduces performance by max. 5-10% (for Cp this will be less).