I would support what Kal and Jerry have said, but would like to add to their comments.
Within the practice of Push Hands, there are variations of training. First, you use pre-arranged patterns of circles. Then you learn to switch spontaneously from one pattern to another. Ultimately, you will be switching among all combinations of horizontal, diagonal, vertical, open, closed, wrist, elbow, and shoulder circles. The open circles can be done with the arms in mirror image (like in Cross Hands) or with the arms offset in time and alternating (like in Cloud Hands). At some point, you will even explore doing one circle with one arm and a different circle with the other, to the extent the various combinations are physically and mentally possible.
Mixed in with all this circling is a study of the Eight Gates. While the circling itself can teach the energies to some extent, in general you must begin to use applications to get a real sense of them.
Within this basic study of Push Hands applications, you are generally prohibited from striking, kicking, or contacting the body below the waist. I think this is partly for theoretical training reasons and partly for safety. I also suspect that there is some variation in the "rules" within Yang Style and definitely between Tai Chi styles.
Within the Eight energies, Ward Off, Rollback, Press, and Push generally employee long energy and can be done relatively safely between relatively untrained partners. Pluck has aspects that can begin to be dangerous, and Elbow, Shouder Strike, and Split have definitely increased risk associated with them. That is because these are generally associated with short energy. You only start to work extensively with these when both partners have more experience and are less likely to injure each other.
Within the Push Hands applications, you work on doing two things: neutralizing and issuing. You do this by using sticking, adhering, connecting and following and by applying the Ten Essentials. Some of the Ten Essentials have a somewhat different range of application to Push Hands than they do to solo form practice (e.g., Distinguish Full and Empty, Seek Stillnes in Movement).
In executing "applications," you have a range of outcomes to train, going from simply neutralizing an opponent's technique, making an opponent unable to apply power to a technique, making an opponent unable to apply power to any technique, making an opponent unable to move, making an opponent have to signal surrender, making an opponent confused, making an opponent unsteady, offsetting an opponent, making an opponent take two steps, making both feet leave the ground, and taking an opponent to the floor.
An essential component of this training is safety; however, this is not because all of the above is inherently safe. This depends on the practitioner. In fact, many applicatons of the Eight Energies can be much more dangerous than a simple punch or kick usually is.
Striking and kicking are not trained in this setting, not only for increased safety, but also for theoretical training reasons. Push Hands is primarily about developing adhering, sticking, connecting, and following. Strikes inevitably involve a degree of disconnection that pushes training in a different direction.
There is, in fact, a rule in some versions of Push Hands training that prohibits striking, but only as long as your partner remains "connected." This is to give a powerful incentive
to train sticking and adhering, and not to train other things. It also helps teach why "not losing" the opponent (Bu4 Diu1) is an important self-defense skill. By the way, by "disconnecting," I am talking about what you do with even one hand, since this can signal either that you are vulnerable to a strike or that you yourself are preparing to strike.
As for Qin Na (Grabbing and Controlling), I would say that this is definitely trained to some extent within Yang Style Push Hands, but there is no suspension of general Tai Chi principles or the orientation of the training regimen. "Controlling" runs the gamut between physically locking joints (usually indirectly) to psychologically making your opponent "Double Weighted."
Some techniques actually require grabbing or joint manipulation. For instance, a one-handed Rollback may well require you to grab the opponent's forearm. To take your opponent to the ground with "Needle at Sea Bottom" will require you to hold the opponent's fingers in place to prevent disconnection.
While grabbing is permitted and occasionally even required, over-reliance on grabbing runs contrary to the goal of learning to stick and adhere merely through contact. You must still "Give up yourself and follow the other" and cannot simply manhandle your opponent at will.
Once you become comfortable with all of the above, including all the basic stepping variations, it begins to make sense to broaden the range of techniques into striking and kicking during free sparring. At this point, one can really talk in depth about Understanding Energy" and using energy naturally. To my understanding, it is not a question of going from "play fighting" to "real fighting" but rather of broadening the range and freedom of techniques.
I should also clarify that you can always explore individual movements (e.g., from the form) outside of Push Hands or sparring, but this is most profitable only when you have sufficient training to use and understand the Tai Chi principles behind them. You can, of course, train "applications" without such an understanding, but then we are talking about training fighting in general, and not about training Taijiquan techniques.
Lastly, I should hasten to say that training Push Hands does not have to be about fight training. Taijiquan has come to be a well that you can drink from at any depth.
I have done Push Hands profitably and safely with elderly women in minimal physical shape and with no experience in contact sports, let alone any martial arts training or inclination. From such sessions, we have both been able to understand Taijiquan a little more deeply. Push Hands can be about learning how to fight or defend yourself, but it can also be about helping each other to deepen understanding of the form, enjoy it more, and get more out of it.