Yes, as mentioned a zillion times, amping helps these a lot. If you get a slightly "warmer" amp, it should help tame the highs a bit as well. The Schiit Vali is generally recommended as a better pairing for the AKG's than the Magni because of this.
Regarding the Modi, I'd say that depends on your sound card. I have found that the difference between the Omni and Modi is pretty subtle. Once I spent a lot of time listening for the differences, I think I trained my ears to hear them better. But it wasn't anything dramatic. I'd almost suggest skipping the Modi if you have a decent sound card and upgrading to something in a higher price bracket.
BTW, if you do have a decent sound card, you're going to want a DAC with optical input rather than USB.
I was more worried about the harshness of the highs when I asked and not so much about the general sq and volume. I had definitely known that I had to be prepared to buy an amp when I ordered the K712. I don't mean to be the type of person that repeats a question that has, in fact, been asked a zillion times .
The audio on my PC is handled by the motherboard. It's a GA-990FXA-UD3 in case you want to take a look at it. It's an okay motherboard and has optical as well as 5.1 capabilities but I don't think it's anything special when it comes to sound quality. That's why I'm not sure as to whether to just go with the Mixamp and an amp or to go with a DAC as well.
As for the Vali vs Magni, I'm hesitant with the Vali because it doesn't work so well with lower ohm headphones (just in case I want to use my M80s or GR07s with it).