It would really depend on whether the antifreeze trait were recessive, dominant, or something in between (incomplete dominance) . In other words, if a plant has one copy of the antifreeze gene, can the plant just use that one recipe more often to make as much antifreeze as it needs or will it make less antifreeze? Will it still make enough to make a difference?

The fragile X thing is a whole different matter. Fragile X is a syndrome (collection of traits) caused by a whole bunch of repetitions (sometimes hundreds) of a DNA sequence in an X chromosome.

You might be thinking about X-linked traits, for which a male only has one copy of any gene on the X chromosome. This does relate to your question. Females have 2 X chromosomes, but in each cell, one X chromosome is randomly inactivated. This means that each cell in a mammalian female really has only one X chromosome, but about half of the cells have one X and the other half have the other X. You can see the resulting pattern in a calico or tortoiseshell cat. In some cells the "orange" code X chromosome is active, while in others the "black" code X chromosome is turned on. In that case it makes a difference whether the cat has two "orange" alleles or only one. However, with colorblindness, females can compensate for having only one copy of the gene that allows color vision by using it more often. So it doesn't matter whether she has one copy or two.

I don't know whether the antifreeze gene can be regulated by the plant to pump out more antifreeze or not. I'd suggest that you try to experiment with this, but strawberries are usually not grown from seeds. They are usually grown from offshoots of plants, so they are actually all clones of the "parent." So you might have trouble doing the cross and trying to grow the plants from seeds. Maybe you can find another plant with the antifreeze gene for an experiment.