How diphenylcarbazide is used to detect Mercury II ions?
The Diphenylcarbazide Test for the presence of mercury was used as early as 1920s. However, since the test is very delicate (the test can detect $\pu{250 \mu g/L}$ mercury in solution), it has been used predominantly in neutral solutions, until it have been adapted to Quantitative Analysis Scheme by Scott in 1929 (Ref.1) to ditect in acidic medium:


  The mercury is precipitated in Group I is dissolved in aqua regia, that of Group II is dissoved in hydrochloric acid and sodium chlorate. In either case, the solution is evaporate to a volume of about $\pu{1 cc.}$, placed in a test tube, and diluted with $5$ to $\pu{6 cc.}$ of water. Four to eight drops of a saturated alcoholic solution of diphenylcarbazide is added and a large excess of solid sodium carbonate is gradually dropped into the solution. When the mercury is present, the form produced on neutralization assumes a blue tinge, and after the addition of an excess of carbonate the entire solution turns blue. 


The author stated at the end that neither sodium acetate nor sodium bicarbonate can be successfully substituted for the sodium carbonate, indicating the sensitivity of the test for $\mathrm{pH}$. It is worth noting that the test outlined here (vide supra) has been successfully used in Quantitative Analysis Courses in University of Georgia, Athens, GA for two years before publication.

The major drawback of the test is the deterioration of stock solution of diphenylcarbazide in different solvents. Thus, it needs to be freshly prepared. To avoid this problem, a solution has been found in the publication by Urone by 1955 (Ref.2). Accordingly, non-aqueous ethyl acetate and acetone were the better solvents, diphenylcarbazide solutions of which are stable for months. Diphenylcarbazide solutions of Methyl ethyl ketone, methyl cellosolve (2-methoxyethanol), and 2-propanol are usable for 1-2 weeks. Aqueous solutions and solvents tending to be basic such as methanol and ethanol, and those containing traces of water and basic impurities, do not make good solvents for stock solutions of the colorimetric reagent (Ref.2).

Notes: (1) All solutions tested (diphenylcarbazide solutions in seven different solvents) in Ref.2 are $0.25\%$ in diphenylcarbazide. (2) Also, you don't have to synthesize 1,5-diphenylcarbazide; it is commercially available in Millipore-Sigma. (3) I don't think anybody can answer your question about "how are the test strips for heavy metals in drinking water made from this compound," because it should be a corporate secret, similar to $\ce{KI}$ papers for oxidant check. 

References:


Alfred W. Scott, “Adaptation of the Diphenylcarbazide Test for Mercury to the Scheme of Qualitative Analysis,” J. Am. Chem. Soc. 1929, 51(11), 3351-3352 (https://doi.org/10.1021/ja01386a503).
P. F. Urone, “Stability of Colorimetric Reagent for Chromium, s-Diphenylcarbazide, in Various Solvents,” Anal. Chem. 1955, 27(8), 1354-1355 (https://doi.org/10.1021/ac60104a048).