So, I have an XX, & I got an X from my mom, & an X from my dad. My dad would have gotten his only X from his mother, & my mother could have gotten the X I got from her father or mother. That would mean I did not get either X gene from either my mom's mom or dad, or my dad's dad - does that mean that I have "less" of their genetics in me? Say I got my mom's dad's X, would that mean I have less of my mom's mom's genetic traits?
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Speaking of your DNA in general, you have alleles from all four of your grandparents. The 22 pairs of autosomes (non-sex chromosomes) can receive DNA from all four with random mixing. On a few chromosome pairs you may be missing alleles from one of your grandparents due to the randomness, but on average you will have about a quarter of your autosomes from each of them.
The X (and Y, for males) chromosomes are special, though. The X you inherit from your mother will have alleles from both your maternal grandparents. The X that you, as a female, get from your father has no alleles from his father, so that grandfather's X alleles are always missing from your genome.
Because of the random mixing, you will have slightly varying amounts of DNA from each of your grandparents: it won't be exactly 25% for each of them. Because of the special X inheritance, your paternal grandfather will on average contribute slightly less than the other three grandparents. (Because of another quirk, mitochondrial inheritance, your maternal grandmother will contribute on average a very small amount more than the other three grandparents.)
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$\begingroup$ Regarding your second paragraph, you should consider recombination between the pseudoautosomal regions of the X and Y chromosome. $\endgroup$ Commented Jun 10, 2015 at 0:56
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$\begingroup$ Nice summary. Sex chromosome inheritance is interesting, for instance that paternal grandmothers are on average more closely related to their granddaughters than their grandsons: Fox et al. 2009. Grandma plays favourites: X-chromosome relatedness and sex-specific childhood mortality. Proc. Roy. Soc. B $\endgroup$ Commented Jun 10, 2015 at 10:52