Coordinating a call or a deadline across a few timezones usually means doing the math more than once. Timezone Converter lets you set one date and time and see it converted everywhere you've added, all at once, sorted by offset.
Because the same instant can fall on a different calendar date depending on the timezone. If it's 11pm in your base timezone, several others may already be in the next day, or still in the previous one; the badge makes that immediately clear instead of leaving you to work it out.
Yes. Offsets and converted times are calculated for the exact date you choose, using each timezone's real rules, including Daylight Saving transitions, so a date in July and a date in January for the same city can correctly show different offsets.
The timezone database itself only names one representative city per zone, so, for example, Mumbai, Delhi, and Kolkata all share a single "Asia/Kolkata" entry. Searching and adding by the cities people actually recognize is easier, even when several share the same underlying zone.
No. All of the time math happens locally in your browser. Nothing about what you're converting is sent to a server.