Where you seem to be confused is the idea that a FF lens collects more light than an APS-C lens with the same focal length and aperture. Whether this is true or not depends on exactly how one defines more. In terms of total photons collected the FF lens does collect more, but it does so only because it collects them from a wider angle of view and projects them onto a larger image circle. Exposure, however, is not expressed in terms of total photons collected. It is expressed in terms of photons collected per unit area of the sensor or film. If an image circle covers twice the area of the image circle of another lens, the lens must collect twice as many photons to produce the same number of photons per mm².
Does the aperture on the full frame lens compare directly to the aperture on the APS-C lenses, or will it be a wider aperture with respect to my smaller sensor?
The aperture on the full frame Zeiss Sonnar 35mm f/2.8 is not wider than the f/1.8 apertures of the other two lenses - it is narrower by just over one stop.
Regardless of the size of the sensor behind it, the f-number is determined by the focal length of the lens divided by the size of the entrance pupil (the aperture as seen through the front of the lens whose apparent size is affected by the magnification of any lens elements located between the front of the lens and the physical aperture diaphragm). A FF 35mm f/2.8 lens does not collect more light than an APS-C 35mm f/2.8 lens because the entrance pupil is larger - the entrance pupils are the same size for the same f-number. The FF lens collects more light because it collects light from a wider angle of view than the APS-C lens with the same focal length does. The amount of light each lens collects from any particular spot in the field of view shared by the two lenses will be the same, it's just that the FF lens collects more of the scene and projects it in a larger image circle than the APS-C lens.
When mounted on an APS-C camera, a 35mm f/2.8 lens that casts an image circle large enough for a FF sensor will act exactly like a 35mm f/2.8 lens that only casts an image circle large enough for the APS-C sensor. The additional parts of the image circle cast by the FF lens aren't recorded by the sensor and have no effect on the photograph captured by the smaller sensor. When we say depth of field is affected by the differences between an APS-C camera and a full frame camera it is the differences in the sensor sizes that we are comparing, not the differences in two lenses with the same focal length and apertures used on the same APS-C camera. How the DoF is affected is determined by a large number of variables.
There are so many variables to deal with in this question and most of the answers assume several without specifying those assumptions. This leads to gross misunderstandings about the relationship of focal length, aperture, sensor size, shooting distance, display size, viewing distance, and even the visual acuity of the viewer to Depth of Field (DoF). All of these factors combined will determine the Depth of Field of an image. This is because DoF is a perception of what range of distances from the focal plane are in focus. Only one distance from the focal plane is actually in focus such that a point light source will theoretically produce a point of light on the focal plane. Point light sources at all other distances produce a blur circle that varies in size based on their proportional distance to the focal plane as compared to the focus distance. DoF is defined as the range between the near and far distance from the focal plane that the blur circle is still perceived as a point by the viewer of an image.
We ask questions such as, "How does depth of field change when using the same lens on a camera with a different sized sensor?" The correct answer is, "It depends." It depends on whether you shoot from the same distance (and thus change the framing of the subject) or shoot from a difference distance to approximate the same framing of the subject. It depends on whether the display size of the image is the same or the display size of the image is changed by the same proportion as the different sensor sizes. It depends on what changes and what stays the same in regard to all of the factors cited above.
If the same focal length is used at the same subject distance with the same aperture using the same sensor size with the same pixel density and printed at the same resolution on the same size paper and viewed by persons with the same visual acuity then the DoF of the two images will be the same. If any one of these variables change without a corresponding change to the others, the DoF will also be changed.
Please note that displaying an image from an APS-C camera at the same display size as an image from a FF camera requires greater enlargement. This also enlarges the blur circles in the image and necessitates using a smaller circle of confusion when calculating DoF. But in the case of using two different lenses on the same APS-C camera, the enlargement ratio does not change so neither does the DoF. If both lenses have the same focal length and aperture and were shot from the same subject distance by the same camera then the DoF will be the same (as long as both are used within the comparative resolution limits of the two lenses).
Depth of field isn't really affected by whether the lens projects a FF or APS-C sized image circle per se. Rather, it is affected because the image collected by an APS-C sized sensor must be enlarged by a greater factor to be displayed at the same size as an image collected by a FF sensor. The more you enlarge the image projected by a lens onto a sensor, the more you enlarge the blur. What we call Depth of Field is just a way of expressing the amount of blur we consider acceptable because it will still be perceived as sharp at a specific enlargement ratio and viewing distance.
Are there any other concerns I should have about using this full frame lens on my smaller camera?
Typically full frame lenses of the same focal length and maximum aperture are larger, heavier, and more expensive than their APS-C counterparts. This is due to the need for the rear of the lens to project a larger image circle. Any other differences between two different lens designs must be compared on a case-by-case basis of the particular lenses in question.