Make "Countdown" aware of time locales

Summary:
When you type "5 PM", it should mean "5 PM in your local time", not "5 PM in the
server's time". Use locale-aware functions to read and write times.

Also a couple of usability tweaks.

Test Plan:
Swapped my time between America/Los_Angeles and America/New_York and created a
bunch of countdowns.

Reviewed By: jungejason
Reviewers: hsb, toulouse, jungejason, tuomaspelkonen, aran
CC: aran, jungejason
Differential Revision: 531