Yes, a US permanent resident needs an Electronic Travel Authorization (eTA) to visit Canada.
Yes, but you have to have your passport
Yes, a US resident can travel to Canada.
Yes, a US permanent resident can work in Canada with the appropriate work permit or visa.
Yes, a US permanent resident can travel to Japan. They will need to obtain a visa before traveling to Japan.
Yes if you have a Permanent Resident card; however, if you do not have a Permanent Resident card you would need to apply for a visa in advance.
Yes you can.
http://wiki.answers.com/Q/How_soon_you_can_travel_after_becoming_a_permanent_resident"
Yes, a US permanent resident can travel to Colombia. They will need a valid passport and may also need a visa depending on their country of citizenship. It is recommended to check the specific entry requirements before traveling.
The US has a visa waiver program under which nationals of selected countries are allowed to enter the US for periods of less than 90 days without a visa (you still need a passport) provided they meet certain other criteria as well. Permanent resident status does not make you a "national" of the UK; what matters is where your citizenship is. See the link in the related links section for more information. No idea about Canada.
Hawai'i is a US state, therefore no passport is needed for US citizens to travel to or from Hawai'i. Hawaii is part of the United States. Therefore, if you are a Citizen or a Permanent Resident(Green Card holder) you won't be needing a passport to go there. DL will do.
Assuming you're not a permanent resident in Canada - you would file a USA tax statement.