What is the best short (i.e., roughly 20 item) measure of Big Five personality? What is the best short set of 20 questions that can give meaningful information about a person's big five personality? In case that's not possible with 20 questions, what's the minimum and where do I find such a set of questions?
In most cases, increasing the number of items on a personality test will increase the reliability of measurement (see the Spearman-Brown formula). Thus, there is a trade-off between test length and reliability. 

There are several 10 item measures of the Big 5. Notably, the TIPI and BFI-10 (Rammstedt et al, 2007). However, I would strongly discourage anyone from using a 10 item measure unless you are severely limited in time.

My feeling is that around 10 items per measure (i.e., 50 items in total) starts to give you quite good reliability and allows for breadth in item measurement. Many standard measures of the Big 5 that are not trying to measures facets within the Big 5 are around this length (e.g., NEO-FFI, IPIP 50 item, etc.), although 100 items will of course have benefits for reliability. See this answer for suggestions of additional scales.

So in some respects 20 to 25 items is a middle ground between too short and a good length. I would only use such a scale if time constraints were a major factor.
You might want to have a look at the Mini-IPIP (Donnelan et al 2006). It's a 20 item measure based on the IPIP.

References


Donnellan, M.B., Oswald, F.L., Baird, B.M., &amp; Lucas, R.E. (2006). The mini-IPIP scales: Tiny-yet-effective measures of the Big Five factors of personality. Psychological Assessment, 18, 192-203.
Gosling, S. D., Rentfrow, P. J., &amp; Swann, W. B. (2003). A very brief measure of the big-five personality domains. Journal of Research in Personality, 37(6), 504-528. doi: 10.1016/S0092-6566(03)00046-1
Rammstedt, B., &amp; John, O. P. (2007). Measuring personality in one minute or less: A 10-item short version of the Big Five Inventory in English and German. Journal of Research in Personality, 41(1), 203–212. Available online, URL: http://www.westmont.edu/_academics/departments/psychology/documents/Rammstedt_and_John.pdf.