Is it possible for a singular planet to rotate around a single sun in the same way that our moon rotates around the Earth? If so, would it be possible for that planet, if as large or larger than Earth, to have gravity?
Yes. Literally everything that has mass (and even some things that don’t) will “have gravity”. It is certainly possible for a single planet to orbit a single star (as opposed to our planet, for example, which is accompanied by the other planets of the solar system). In fact, if anything, a single planet will be more stable because it will not be perturbed by other massive bodies. But in all of these cases, gravity will certainly be present. For example, the reason a planet will even stick together to be a planet is because gravity holds it together.
Responding directly: Yes it is possible and is expected in cosmological time scale.
It is expected for both bodies to have the same face to the other with sufficient time, but the one of smaller mass will generally slow its rotation with respect to the body of major mass first, as strictly that phenomenon is connected to the angular moment of rotation, that moment is proportional to mass and speed of rotation, but other factors influence the loss of rotation (braking), such as the existence of fluids in the celestial body (plasma in the stars and atmosphere and water or other fluid in the planets and satellites, the denser the braking of the celestial body, the energy in this case becomes tidal energy in the fluids. What speeds up the process is proximity. The body that will first present the same face is the one that has the lowest energy (moment) and the set of conditions that predispose them to lose energy faster, such as less mass, lack of atmosphere and oceans.
The earth does not yet have the same face for the Moon, but millions of years ago the day on earth lasted only about 10 hours, from here a few million years ago when the day on earth lasts for a lunar month, both the Moon and Earth will show the same face and will be synchronized. In his question the case is of the existence of a single planet in the system, this only facilitates the synchronization because there is less gravitational perturbation of other celestial bodies of great mass, but remembering that the proximity is the main factor in the speed of the process.
See Wikipedia article: Gliese 581c
Search the internet for these titles for more information: Tidal coupling (Tidal Locking)
You will see that the Pluto system and its moon Charon are already synchronized each showing the same face to the other component of the system and there is a planet in the star Tau Bootes that presents this synchronization.
There is also an excellent article on the subject in the book "Asimov explains" by Isaac Asimov
Answering the second question:
Yes, the planet independent of size will have gravity, but the resulting value will be different on each side. On the side permanently presented to the star will be smaller than on the other side but will not be zero. There will be a point between the star and the planet where gravity will vanish, but it will still be off the planet. The loss of momentum of rotation partly alters the moment of translation, perhaps making it happen to the planet what already happens to our Moon that moves away from Earth at a speed of 3.78 centimeters per year, with the possibility of tearing from the earth.
Excuse my english please.