First off, let me point out that if you have a central planet which is orbited by other bodies, those other bodies must necessarily be considered moons and cannot be planets. This is due to the official definition of a planet stating that it must orbit the central star. The objects orbiting your central planet are not primarily orbiting the central star and thus would be considered moons or satellites.
What you're proposing is that your central planet is orbited by two moons that are on opposite sides of the planet. This condition provides some strong constraints on your system as dictated by the laws of physics.
If we are to consider that the moons are orbiting around the central planet, then the planet must be fairly massive. If all three of your bodies are of a similar or comparable mass, you'd have a much different (and not necessarily stable) type of orbit than what you're describing (see this link for example). The fact that you have two moons orbiting your planet, without your planet having a noticeable orbit about the moons, implies the planet has most of the mass of the system. This results in one of two scenarios, either
- your central planet is very large (e.g, the size of Jupiter or Saturn) with moons the size of Mercury or Ganymede, or
- your central planet is relatively small (e.g., the size of Earth or Venus) with moons the size of asteroids (e.g., Phobos and Deimos, the moons of Mars)
Of course you could have any gradation between the two possibilities I just listed. The main point is, your planet must be much more massive than your moons in order to be the dominant gravitational player in its system.
You also state your moons are orbiting such that they're on opposite sides of the planet. Such an orbital configuration is known as a 1:1 resonance. It necessarily implies that your moons are orbiting at the exact same distance from the planet and take the same amount of time to complete a single orbit. While such a system is technically stable, I can't say with certainty that it is stable for a long period of time. More than likely, small perturbations will occur that will build up over time and cause your moons to leave their 1:1 orbital resonance. This will result in unpredictable outcomes, but my best guess is that the moons may start gravitationally interacting if they get close enough after no longer being on the same side of the planet. Eventually this may result in them exchanging energy and one getting kicked to a closer orbit while another getting kicked to a further orbit. But that's just a guess.
So short answer, yes your system is possible but it comes with some constraints and is likely not going to be stable for a long time period.
You also ask about the habitability of such a system. This is a bit tricky and there's no single answer. There are so many conditions that affect habitability. To make matters worse, we only have a single example of a habitable planet. However, there are general conditions we believe must be true for a planet to be habitable (for humans at least).
The big condition is that your system must be at a good temperature such that water can exist as a liquid. If your central planet is a Jupiter/Saturn analog, this is not going to happen since such planets have large, mostly hydrogen gaseous envelopes. There's not really any pools of water for humans. However, your moons could still host water. It really depends on how far out your planet is from the star. These types of planets form very far from the star, where water doesn't easily exist as a liquid so either your planet must migrate in to the inner stellar system early on (and stop at a respectable distance), or else some sort of tidal effect has to keep your moons much warmer than they would be other wise (see Enceladus, for example). The former means your chosen moon orbits are likely to be less stable and the latter means you're going to need other moons to help provide the heating, again making your orbits less stable. If your central planet is Earth-like, it can certainly host water and be as Earth-like as you want, but the moons are unlikely to be big enough to be habitable.
There are various other conditions for habitability such as nice atmospheres or magnetospheric protection from the star but one could write a whole book on this subject. I'll leave off by pointing you to this question on habitability.