Using a t-test for paired proportions This is something that I'm having trouble verifying on Google, so I'm asking here as a sanity check.
I have data collected randomly (not equally) over time for different users. To determine if some variable has an effect on user's behavior, I'm looking at percentages of counts of the user visiting a certain place (place A) within and outside a certain time window.
Here's an example of what the data might look like:
test1 = rbinom(30, 14, .27)
test1 = test1/max(test1)

test2 = rbinom(30,10,.3)
test2=test2/max(test2)

So test1 would be the percent of times users went to place A outside of a certain time window, and test2 would be the percent of times users go to place A inside the time window. 
Would it make sense to use a paired t-test to see if users go to place A more often in this time window?
t.test(test2, test1, alternative="greater", paired = T)

 A: It is unclear whether the two proportions have to sum to 1. If they do, then you only need to look at one of them, because the other is redundant. However, the code supplied assumes they do not. In fact, for your code, the total of test 1 and test 2 is often above 1.
I think your code has some confusion about the denominator. Using the same variable name twice doesn't help this, so let's change it to:
test1 = rbinom(30, 14, .27)
prop1 = test1/max(test1)

test2 = rbinom(30,10,.3)
prop2=test2/max(test2)

test1
test2

These are clearly counts. A t-test is really for continuous data. If you want to change these to proportions, then the denominator should not be max(test1) but the total number of times, which would be 14 for test1 and 10 for test2. If you did that, then a paired t-test would make sense:
t.test(prop1, prop2, alternative = "two.sided", paired = TRUE)

A: Chi-square goodness of fit test is appropriate to answer this question. Paired t-test would be a more intuitive choice if you had absolute numbers of counts, rather than proportions summing to 1, and if you were more implicitly interested in the difference of means rather than distributions between the two categories, but I guess depending on how you approach this, the difference can be subtle. 
Note that the example you provided does not generate the same dataset as implied in the formulation above, where presumably the two percentages some up to unity, whereas a pair of random numbers is unlikely to do so (or else clarify the question).
A: The t-test formally assumes normally distributed data so may look problematic, however the binomial(n,p) distribution is very well approximated by the normal distribution unless the n is very small or the p is very close to zero or one. So in the binomial setup that you're simulating, the t-test looks fine by me, all the more so because I don't know a standard alternative for paired proportion or count data (although there may be one flying around somewhere in the literature).
Whether you divide by the max to make proportions out of counts is actually irrelevant for the applicability of the t-test (although it is relevant for the result because the interpretation of what you are testing changes if you divide both counts by different numbers). The data is discrete anyway (which isn't a problem because of good normal approximation), it only changes the variance.
Your real data problem may be slightly different than the binomial example because you probably won't be able to justify a binomial number of trials n. A Poisson distribution may be better as a model, but there may be over- or underdispersion. Once more I'm not sure whether a paired test for such data exists (one could google for Poisson paired test), however I would expect the t-test to work well unless your distribution is very skew. Note that for the t-test to be approximately correct it is not even necessary that the underlying distribution is approximately normal, it suffices that your number of observations is large enough as long as your distribution of differences between the paired values isn't strikingly skew or has extreme outliers, because of the Central Limit Theorem.   
