Hezbollah's primary areas of activity are Lebanon and Syria.
Some of Hezbollah's senior leadership live in Lebanon and the remainder live in Syria and Iran.
Lebanon
Hezbollah is located all over Lebanon, once a jewel of the Middle East, now a devastated wasteland thanks to Hezbollah.
Syria removed all of it's military troops from Lebanon.
Syria removed all of it's military troops from Lebanon.
Yes and Hezbollah won and Israel asked for cease fire from Hazbollah.
Hezbollah is not a country and does not, therefore, have a capital. Hezbollah's operations are primarily based out of Lebanon, but are scattered throughout the country in order to avoid Israeli countrerstrikes and assassinations.
Hamas in the Gaza strip Hezbollah in Lebanon
The Lebanese support for Assad primarily comes from Hezbollah control in the government. If Sunnis and Christians in Lebanon were able to strengthen the national army and oust Hezbollah from government power (this is nearly impossible because Hezbollah has always been more powerful than the national Lebanese Army), the position of Lebanon would probably move closer to neutrality.
There was no parent organization for Hezbollah. It grew out of the need for Shiite Muslims to advocate their interests militarily in response to Christian Falangists and Sunni Extremists in Lebanon (who fought militarily).
There were a few wars that Israel fought in Lebanon: the Lebanese Civil War from 1981-1982, the Occupation of Southern Lebanon from 1982-2000, and the War Against Hezbollah in 2006.