Transformers are inductors. They just happen to have mutual inductance with another inductor. Like inductors, they also have self-capacitance, and so a resonant frequency. They also have loss, and consequently some maximum power handling ability which if exceeded causes the transformer to fail by overheating.
As such, any parameter which is relevant to an inductor might also be relevant to a transformer. There is no general way to say what parameters are relevant or not: to answer that question requires some understanding of the circuit in question, and its operation and requirements.
For example, you'll need to consider the magnetizing inductance of the transformer. This is the inductance that the transmitter sees looking into the transformer as a result of the transformer itself. You can measure it by leaving the secondary disconnected and measuring the transformer as an inductor.
In your end-fed dipole example, this magnetizing inductance appears in parallel with the antenna. There are two ways to approach this problem.
One way is to make sure the magnetizing inductance is an impedance much greater than 50Ω (50Ω being, hopefully, the impedance of the antenna, as seen by the transmitter). This will mean that the magnetizing inductance has negligible effect on the overall impedance seen by the transmitter. (If that doesn't make sense, consider the impedance of a 1kΩ resistor: what happens if you add a 1MΩ resistor in parallel?)
If it's not practical to obtain a sufficiently high inductance, then a capacitor can be added in parallel with the transformer to make a parallel LC circuit, which at resonance has a very high impedance. The interwinding capacitance of the transformer also effectively contributes parallel capacitance, so it's possible to design a transformer which is self-resonant and does not require an additional capacitor.
Another way to deal with the magnetizing inductance is to incorporate it into the design of the circuit. If you needed an inductor anyway, perhaps as part of a filter or a matching network, then you can put the primary of an appropriately designed transformer there instead.
So what does this mean for your end-fed dipole example? What things might we change, and what effect would they have in that application?
We could increase the number of turns, while maintaining the same turns ratio. This will increase magnetizing inductance, which might be good since we want that impedance to be high. More turns will also increase the interwinding capacitance. But more turns will also increase the magnetic flux through the core, eventually hitting the limit of saturation.
We could use a different material. But we'd want to consider the losses inherent in that material, its permeability and the effect that will have on the magnetizing inductance, and its maximum flux density and how that will impact the peak current the transformer can handle.
The physical size of the core is also a concern. A larger core spreads the magnetic flux through it through more area, and so can handle more magnetic flux (and consequently, current and power) before reaching saturation. A larger core also has more area to lose heat from losses to its surroundings, and more thermal mass to absorb transient power peaks, and so can handle more power.