Does it handle daylight saving time?

Yes. Times are calculated for the exact date you choose, using each city's rules, so clock changes are included.

How do I find a time that suits everyone?

Add each person's city. Hours shaded green are inside everyone's working day; the planner highlights the best overlap.

Yes. Choose a time and use Add to calendar to get an invite file that Apple Calendar, Google Calendar and Outlook can open.

Your own time zone is added automatically. Search for the other cities and add them.
Pick the date. Clock changes for daylight saving are handled for that exact day.
Tap any hour. The summary shows the time in every city, with a note when it falls on a different day.
Share the times as text, or use Add to calendar for an invite that opens in Apple Calendar, Google Calendar or Outlook.

Always write the time zone. "3pm" means nothing to a team in three countries. Share the list of local times.
Mind the clock changes. Europe and North America change on different weekends in spring and autumn, so overlaps shift for a week or two. Check the actual date.
Rotate the pain. If one person always takes the 7am or 9pm slot, alternate who gets the awkward hour.
Use the overlap for conversation, and do the rest asynchronously.

Cities	Typical shared working hours
London and New York	About 1pm–5pm London, 8am–noon New York
London and San Francisco	Little or none; usually 4pm–6pm London, 8am–10am San Francisco
New York and Tokyo	Early evening New York is early morning Tokyo, next day
London and Sydney	Early morning London is evening in Sydney

These shift with daylight saving; the planner shows the exact hours for your date.

Yes, in this browser. Nothing is sent anywhere.

India, parts of Australia, Newfoundland, Nepal and others are shown with their exact minutes.