Your browser already knows the rules for every time zone, including daylight saving time changes. This page takes the date and time you enter in the chosen zone, works out the exact moment in UTC, and shows what the clock reads at that moment in each city below. Nothing is sent anywhere.
Yes. Offsets are looked up for the date you pick, so a meeting in March or November shows the correct difference even when the US and Europe change clocks on different weeks.
When the other city is many hours ahead or behind, the local calendar date can be the next or the previous day. The date is shown next to each time.
See also: Unix Timestamp Converter · Hours Calculator · Age Calculator