To be fair, one does not need to go ass to grass on squats. Parallel is more than fine. It depends on the persons mobility, goals, what they are working on, etc. I myself decided to reset my squats and am starting at 225 lbs doing 4x6 @ 60-75 seconds rest between sets. Really want to focus on my glute activation and am going as deep was I can.
It also depends on the type of squat. On low bar squats just a hair below parallel is fine, since going ass to grass will not only kill hip drive, but will potentially end up in a curved lower back since the knees have traveled forward enough to pull the hips in, resulting in lumbar flexion. ATG should be reserved for Front and High Bar Squats.
But yeah, if you are not a the very least hitting parallel (ie, the femurs are parallel to the ground), it doesn't count.
Yes, during the squat you should aim to "bounce" out of the bottom. Rippetoe goes on excrutiating detail about this in SS but the gist off it is that there is huge difference between how negative-positive (squat, bench press, dips) and positive-negative (deadlift, power clean, Chinups/pullups) movement works. In the first ones, specially the squat, bounce is expected and sometimes encouraged. In the latter, Bounce is discouraged. A straight up Chinups is better than a kipping Chinup for example for developing strength.
The overhead press is a special case where both techniques work the same and it depends on the lifter.
Well, we'd need to be clear about what Rippetoe is talking about when he's talking about the bounce. The "good bounce" is the stretch reflex, which is the product of releasing the kinetic energy stored in the viscoelastic tissue of muscle fibers and tendons during the eccentric phase of the movement. It works basically like spring, or a rubber band, and the longer the trip (during the eccentric) the tenser the tissue will be and the more energy will be stored to be released on the concentric. This is why Rippetoe advocates the low bar squat, because the angle of the hip ensures that, below parallel, the hamstrings will be at their most tense and that's where the bounce comes into play: you literally bounce off your hamstrings. If you go to low on this type of squat, the hamstrings will slacken and there'll be no way of using the stretch reflex to bounce out.
So in that that sense, any type of bounce that happens as a result of the release of this stored energy during the eccentric will be acceptable, even in stuff like chinups, because there will be stored energy in the triceps, the delts and even the pecs at the bottom. From SS:
"For a high-rep (Chin-up) set, you can use a stretch reflex at the bottom as long as the bottom is actually The Bottom."
Rippetoe, Mark (2012-01-13 00:00:00+11:00). Starting Strength (Kindle Locations 6718-6719). The Aasgaard Company. Kindle Edition.
What is NOT acceptable is kipping, but kipping is very different from using the stretch reflex out of the bottom. Kipping is basically using forward momentum, like a pendulum, to swing up and down. That's cheating, and it leads only to shoulder injuries, not to the development of strength. Crossfitters kip their pull-ups so they can brag they can do a ton of them for time, but since we take strength training seriously, we don't kip. We do bounce out of the bottom though.
The deadlift, due to its particular nature (it starts with the concentric and ends with the eccentric) has no stretch reflex whatsoever. That's the key difference between conventional deadlifts and Romanian Deadlifts. Since Romanian deadlifts do start from the top, there's a stretch reflex from the hamstrings (if done correctly) at the bottom to bounce up again. In fact that is the reason we do RDLs. But in conventional deadlift, there's no stretch reflex because there's no eccentric; the lift is basically over at the top.
Bouncing the plates off the ground is a completely different type of bounce that doesn't involve the stretch reflex whatsoever. Like kipping, is mostly a form of cheating. That's the "bad bounce".
In the bench press, we WANT to use the stretch reflex to bounce of the lats and the biceps which accumulate viscoelastic energy during the eccentric. We don't want to bounce the bar off the ribcage, which would be another example of the "bad" bounce.