Common practise is to compare p-value with three levels - 0.05, 0.01 and 0.001. Since your p-value is less than each of them, you have to choose the smallest one, so you should conclude that differences are significant and p<0.001. Roughly speaking: The smaller the p-value, the more significant differences are.
Since we do not know distribution of your data, we do not also know which test should you use. But you have quite large sample, so there is high chance that parametric test can be appropriate (t-test for paired data).
(Technically, the P-value is the probability of observing data at least as extreme as that actually observed, given the null hypothesis.)
Q1. A decision to reject the null hypothesis on the basis of a small P-value typically depends on 'Fisher's disjunction': Either a rare event has happened or the null hypothesis is false. In effect, it is rarity of the event is what the P-value tells you rather than the probability that the null is false.
The probability that the null is false can be obtained from the experimental data only by way of Bayes' theorem, which requires specification of the 'prior' probability of the null hypothesis (presumably what Gill is referring to as "marginal distributions").
Q2. This part of your question is much harder than it might seem. There is a great deal of confusion regarding P-values and error rates which is, presumably, what Gill is referring to with "but is typically treated as such." The combination of Fisherian P-values with Neyman-Pearsonian error rates has been called an incoherent mishmash, and it is unfortunately very widespread. No short answer is going to be completely adequate here, but I can point you to a couple of good papers (yes, one is mine). Both will help you make sense of the Gill paper.
Hurlbert, S., & Lombardi, C. (2009). Final collapse of the Neyman-Pearson decision theoretic framework and rise of the neoFisherian. Annales Zoologici Fennici, 46(5), 311–349. (Link to paper)
Lew, M. J. (2012). Bad statistical practice in pharmacology (and other basic biomedical disciplines): you probably don't know P. British Journal of Pharmacology, 166(5), 1559–1567. doi:10.1111/j.1476-5381.2012.01931.x (Link to paper)
Best Answer
There are two issues here:
1) If you're doing a formal hypothesis test (and if you're going as far as quoting a p-value in my book you already are), what is the formal rejection rule?
When comparing test statistics to critical values, the critical value is in the rejection region. While this formality doesn't matter much when everything is continuous, it does matter when the distribution of the test statistic is discrete.
Correspondingly, when comparing p-values and significance levels, the rule is:
Please note that, even if you rounded your p-value up to 0.05, indeed even if the $p$ value was exactly 0.05, formally, you should still reject.
2) In terms of 'what is our p-value telling us', then assuming you can even interpret a p-value as 'evidence against the null' (let's say that opinion on that is somewhat divided), 0.0499 and 0.0501 are not really saying different things about the data (effect sizes would tend to be almost identical).
My suggestion would be to (1) formally reject the null, and perhaps point out that even if it were exactly 0.05 it should still be rejected; (2) note that there's nothing particularly special about $\alpha = 0.05$ and it's very close to that borderline -- even a slightly smaller significance threshold would not lead to rejection.