Is this a valid way to think about p-values? Would it be accurate to say that a p-value is a random variable whose null distribution is Unif$(0,1)$ which stochastically dominates its distribution under the alternative hypothesis? I realized that I have been thinking about p-values in the sense of "if we set significance level $\alpha$, then we expect $\alpha$ proportion of research papers to be false positives" which I now think is wrong. I'm trying to readjust my idea of what a p-value really is and I haven't seen it phrased the way I put it so I'd like to know if it's correct. 
 A: $p$ can indeed be regarded as a random variable (in fact, it's a statistic), and is required to be uniformity distributed on $[0, 1]$ under the null hypothesis. However, no guarantees are made about the distribution of $p$ under the alternative hypothesis. This makes sense when you consider that the alternative hypothesis is, for the typical two-tailed test, extremely weak, so very little can be inferred from it.
A: 
Would it be accurate to say that a p-value is a random variable whose null distribution is Unif$(0,1)$ which stochastically dominates its distribution under the alternative hypothesis? 

No, that dominance would be a desirable property, not a definition. Many good (and widely used) tests are biased under some alternatives (it's typical for goodness of fit tests for example). Further, one can easily define "useless" test statistics which don't possess this property anywhere under the alternative. 

I realized that I have been thinking about p-values in the sense of "if we set significance level $\alpha$, then we expect $\alpha$ proportion of research papers to be false positives" which I now think is wrong. 

It is wrong for several reasons. Perhaps the most obvious is that almost all point nulls are actually false (and most published hypothesis tests use point nulls). Consequently while rejections are very common in published papers (in some areas, unfortunately, even ubiquitious) few of them will actually be false rejections even in cases where the hypothesis is not rejected in a later attempt to replicate the result.
A: There have been many different definitions of P-values and there will, no doubt, be many more. I do not find yours to be satisfactory because the decision theory-based concept of stochastic dominance seems to fit more with the dichotomous hypothesis test framework than with the significance testing framework that yields a P-value. The hypothesis testing framework utilises rejection regions and the $\alpha$ that you mention in your question, and does not entail calculation of a P-value at all.
See these questions and answers for more on this topic:
What is the difference between "testing of hypothesis" and "test of significance"?
Why are lower p-values not more evidence against the null? Arguments from Johansson 2011
Is it fair to say p-values tell us nothing about the probability null hypotheses are true?
Is the "hybrid" between Fisher and Neyman-Pearson approaches to statistical testing really an "incoherent mishmash"?
Interpretation of p-value in hypothesis testing
