The "eternal light" in the synagogue is not a permanent yahrtzeit light - it is a sign of the eternal covenant between G-d and the Jews.
If you light the yahrzeit candle before it's time to light the Hanukkah-menorah, you may light the yahrzeit candle first. Otherwise, once it's past sunset, you should wait until about 20 minutes after sunset, light the Hanukkah-menorah, and afterwards light the yahrzeit candle.
Yes.
Yes, I think so.
They throw it away.
Yahrzeit candles are memorial candles that are lit in Jewish tradition to honor the memory of a deceased relative on the anniversary of their death and other significant days. The candle typically burns for 24 hours as a way to commemorate and remember the individual who has passed away.
The Yahrzeit does not come straight after the death, but on the anniversary of the death. Therefore, if you light a candle - even one intended for the Yahrzeit that burns for 24 hours - on the evening after the death has occurred, it is not technically a Yahrzeit candle. I can't see that there would be anything halakhically (by Jewish law) wrong with doing so, but I would most definitely urge you to check this with a rabbi and don't just take my word for it. There are several online "ask the rabbi" services who will be able to help you with this - I'd recommend the one at www.chabad.org as they've never let me down yet and will send a personal reply usually within a day or two.
A Yahrzeit Candle is a special memorial candle. Special candles that burn for 24 hours are used.
The yahrzeit candle burns during the entire roughly-24-hour date during which thedeceased individual is memorialized ... from the sunset that begins that day until thesunset that ends it. When the date happens to be a Sabbath, then the yahrzeitcandle would be lit immediately before lighting the Sabbath candles on Friday night.When the date is a Sunday, then the yahrzeit candle would be lit immediately afterthe recitation of Havdalah on Saturday night.
No prayer is required when extinguishing a Yahrtzeit candle, because according to tradition, a Yahrtzeit candle is not to be extinguished. It should be let burn for the entire 24 hours of the anniversary, beginning at sundown. If the candle must be extinguised because of a safety issue, it may be extinguished without a prayer. If at all possible, however, you should arrange to be around long enough to allow it to burn out naturally. If this is impossible for you, there are alternatives such as electric candles which may be safely left on for 24 hours. There is nothing special about the candle itself, and any candle or light source that can remain lit for the full 24 hours may be utilized. The same light should not, however, then be used for other purposes.
A votive candle or votive lamp.
Yahrzeit is a commemoration of the death of a Jew by a mourner (the child, sibling, spouse or parent of the deceased). The date of the Yahrzeit, which is calculated according to the Hebrew calendar, is the anniversary of the death, not the burial. The anniversary of the death of a loved one is naturally a solemn day, and Judaism helps the mourner experience this pain and and also honors the memory of the deceased via Yahrzeit rituals.
The candle lighting usually takes place in the home. In the synagogue, special prayers are added to the usual daily services.