1- Is one of these statistics (KDE's p-value, KS statistic or the two-tailed p-value) recommended for my needs? If so, why?
Your needs as expressed do not seem to be sufficiently clearly defined as to differentiate between them. They both test for a difference in distribution.
2- What is the difference between the "KS statistic" and a "two-tailed p-value"?
The two sample Kolmogorov-Smirnov statistic is the largest difference in ECDFs for the two samples:
(The data here is the same data I generated for your other question. Here the A sample is red and the B sample is blue.)
The height difference in ECDFs at x=35 is 1/6 or about 0.1667, the same as the value produced by calculating the statistic:
> ks.test(A,B)
Two-sample Kolmogorov-Smirnov test
data: A and B
D = 0.1667, p-value = 0.6228
alternative hypothesis: two-sided
The meaning of the p-value is as for any hypothesis test - the probability of obtaining a statistic at least as unusual (in this case, at least as large) if the null hypothesis were true.
3- Will the difference in the number of elements in each set affect the outcome of these statistics?
No, the KS test, and (to my understanding) the KDE-based one both handle different sample sizes.
All that said, I recommend you consider whuber's words most carefully. There's a lot of good advice packed into very few words.