I was wondering if there is an established method to keep track of the orbit of an exoplanet assuming we know $a$ - the semi-major axis of the orbit, $e$ - the eccentricity of the orbit, and $i$ - the inclination of the orbit. Can we track its position once a month?
Exoplanets are discovered and observed via a fairly wide range of indirect methods, and those are how we measure the parameters of their orbits. Some methods allow us to get complete determinations of the orbits and the exoplanet parameters, while others only allow us to provide constraints on those parameters. If you want to know where the exoplanet is in its orbit right now, then the answer will depend on what method was used to find it.
Wikipedia has a full page on the various methods used to detect exoplanets, but by a large margin the two main ones are transit photometry and radial-velocity imaging.
In transit photometry, we simply set a camera to stare at the star for years on end, hoping to catch the exoplanet passing between us and the star ("transiting"), at which point we detect a slight dip in the brightness of the star. A series of periodic dips allows us to infer the existence of the planet.
If an exoplanet is detected in this way but not with any other methods, we have no information about where it is except at the times when it transits. We get a good idea about how big the orbit must be, and we know how long we need to wait for another transit, but that's it.
In radial-velocity imaging, we measure the velocity of the star as it wobbles under the gravitational pull of the planet. This is limited in measuring the inclination of the orbit (it can't distinguish between a mid-inclination orbit from a higher-inclination one with a bigger, more massive planet), but it does provide a constant stream of data on the orbit.
As such, the orbit itself is fuzzier, but there is a degree to which you can, at any point, point your telescope at the star and get a direct, relevant signal about how the planet is doing.
For the small fraction of exoplanets where we're able to directly image the exoplanet itself, once the orbit has been established then yes, you can point your telescope at the star at any time and directly track the planet.
That's about as much as you can say at this level of detail -- for more, you should really take a deep dive into how the different detection methods work.
The short answer is yes. When a planet orbits a star then both the star and the planet are on opposite sides of their common center of mass at any point in time. Their distance to the common center of mass is proportional to their relative mass.
To simplify. When the star is in its left most position the planet is in its right most position as viewed from earth. The limit of how accurately you can determine the location of the planet is defined by how accurately you can measure the position of the star.