The Q701 presents bass, but its dialed down relative to the mids and highs. Much as you might wish otherwise, it does not meet your articulated standard for bass. The new, more expensive and limited K 702 65th Annies are a better choice. They have more bass.
All the AKG K/Q 70x cans require an amp. Unamped none will work with set-up #1 above. They might work at least reasonably well with #2. However, the headphone out on receivers made during the last 20 to 25 years have been so watered down so often, only someone with your particular model receiver can give you a reliable answer-provided they also have the can you want tested.
There are a number of decent portable amps within your budget. You might want to check the appreciation thread for the can that you ultimately select to get a good match.
So basically the Q701 is the more "accessible" version of the 70x?
Sorry--I don't understand what you mean.