Wow, you have a really poor history teacher. Considering that Canada is one tenth the size of the United States, our influence on both the US and the World is incredibly out of proportion. It's like an ant being able to lift one hundred times its weight.
Canadian scientific disoveries and engineering inventions include (as a partial list): the AC radio tube, Acetylene, the G-suit worn by fighter pilots, automated mail sorting, basketball, the Creed telegraph system, the electric cooking range, the electron microscope, electric street cars, the gramophone, the heart pacemaker, the hydrofoil boat, IMAX movie system, the insulin process for managing diabetes, JAVA programming, the jetliner, plexiglas, quartz clock, electric prosthetic hands, the television camera (and the CCD), voice radio, the telephone, the telephone handset, variable pitch aircraft propeller, the walkie-talkie, wireless radio, the zipper and the Canadarm (which is the primary workplatform and manipulator mounted on the Space Shuttle and International Space Station). All of these things are original Canadian inventions and backed by patent, you're confused about who is copying who.
There are quite a number of Canadians in the entertainment industry, in fact most movies are filmed in Vancouver, Alberta or Toronto these days and much of popular American culture like Saturday Night Live is produced by Canadians like Lorne Michaels, or their music produced by David Foster. Harold Ramis and Paul Haggis write their most popular moives starring Canadian actors like Michael Cera or Seth Rogan. So really, Canadians have been running American "culture" for years. Another partial list of Canadians in movies: Elisha Cuthbert (The Girl Next Door), Pamela Anderson (Barb Wire, Stripperella), Meg Tilly (The Big Chill, Agnes of God), Jill Hennesey (Crossing Jordan, Nuremberg), Carrie Anne-Moss (The Matrix Trilogy), Mike Myers (Austin Powers), Christopher Plummer (The Insider, Barrymore), Donald Sutherland (The Eagle Has Landed, MASH), Dan Ackroyd (The Blues Brother), Rick Moranis (Little Shop of Horrors), Tommy Chong (Chech and Chong's Up In Smoke), Rae Dawn Chong (Commando), Michael J. FOx (Back To The Futuree Trilogy), Bruce McCulloch (Kids in The Hall), Leslie Nielsen (Police Squard), Tom Green (Freddy Got Fingered), Eugene Levy (American Pie, Best in Show, The Man), Catherine O'Hara (Best In Show, A Mighty Wind), Corey Haim (The Lost Boys), Sarah Chalke, Kevin McDonald, Michael Ironside (Totall Recall), Jim Carrey (Ace Venturra, The Truman Show), Norman Jewison (Director: In The Heat of The Night), Leonard Nimoy (Mr. Spock), William Shatner (Capt. Kirk, Boston Legal), Michael Wincott (Metro, Robin Hood: Prince of Thieves), Samanthat Bee (Daily Show w/ Jon Stewart), Neve Campbell (Scream Trilogy), Megan Follows (Anne of Green Gables), Mia Kirshner (Erotica), Sarah Polley (The Sweet Hereafter, Dawn of The Dead-remake), John Candy (Deceased, The Great Outdoors), Rachel McAdams (Red Eye, The Hot Chick), Paul Haggis (Director: Crash), Sandra Oh, Ryan Gosling, Martin Short (Giminy Glick), Ryan Styles (Who's Line Is It Anyway?), Steve Adams, Richard Archer, Robert Beatty, Michael Cameron, Douglas Dumbrille, Bryen Genesse, Jason Jones, Luke Kirby, Patrick McKenna, Sean O'Neill, Curry Graham, Adam Harrington, Anna Paquin (X-Men, X-Men 2), Matthew Perry (Friends), Justin Priestley, Carly Pope, Oliver Platt (Simon Birch, A Time To Kill), David Cronenberg (Director--The Fly, Crash), Matt Austin, JR Bourne, Hayden Christensen (Shattered Glass), Bruce Greenwood (Thirteen Days; I, Robot), Paul Gross (Men With Brooms), Gordon Pinsent (The Shipping News), Callum Keith Rennie (Memento, eXistenZ), Natasha Henstridge (Species, Species 2),
In terms of the military, Canada first formed a national militia to defend itself from the United States during the American Revolution. This militia fought the United States with the British Army during the War of 1812. Canadians and their allies of the Mohawk nation defeated the American Army at the battle of Montreal and forced the withdrawal of the American Army from British North America (which would become the Dominion of Canada). They also carried out a successful raid with the support of British Marines on Washington DC, which saw the destruction of the original White House.
During World War One, the Canadian Expeditionary Force (later the Canada Corps) won many major battles in France: the Second Battle of Ypres (the first time a colonial army forced a European power to retreat on foreign soil), the Battle of the Somme (earning a reputation as shock troops), Vimy Ridge (and held it until the end of the War, the Germans never tried to take the ridge back), The Second Battle of Passchendaele which decided the Third Battle of Ypres and conducted the One Hundred Day Offensive which broke the Hindenburg Line and advanced to Mons. Canada also entered the war in 1914, three years before the United States.
In the Second World War, Canada fought the Battle of the Atlantic, defending the supply routes from North America to Europe; at the end of the war Canada had the third largest allied navy and nearly 1.1 million men in service (out of a population of 11 million). Canadian soldiers fought so well, that they were assigned their own beach during the Invasion on D-Day, Juno Beach. On D-Day, Canadians achieved all their objectives despite a heavy German defense on, schedule, penetrating further into France than any other allied force (the only allied force to do so). Canada liberated Holland fighting through the flooded coutryside, To this day, the Dutch send thousands of tulips to be planted in Ottawa in recognition Canada's role in their liberation.
Canadians also flew a significant number of the dangerous day time bomber missions over Germany in Lancaster bombers. There is a myth that the United States "protected" Canada from Russion invasion after WW2, while in fact Canadian forces had more combat experience (entering the war in 1939 - not 1942) and had plenty of verteran units and war materials. Canada chose to demilitarize in the post-war period.
In fact, Canada had been a full partner in the Manhattan project and right after the war possessed sufficient nuclear weapons to defend herself (immediately after the war, more than the Russians). Once again, Canada chose to decommision its nuclear arsenal after the war, though Canadian Air units stationed at Laar, Germany during the cold war flew bombers capable of nuclear strikes.
As recently as 2006, JTF2, Canada's Special Forces unit was awarded the Presidential Unit Citation by the United States for relieving elements of the 82nd Airborne "preventing significant loss of life" with deadly accurate sniper fire, with one marksman setting the record for a long distance kill at 2,430 meters (1.509 miles) using a .50 caliber (12.7 mm) McMillan TAC-50 rifle. The world's first special forces unit was a joint Canadian/US unit trained in Canada during WW2.
Canada has never kept a large standing army, but always risen to the occassion when needed. We have better ways of dealing with the unemployed and spending federal money.
We simply have very different cultural priorities that Americans, one of the reasons that the chances of a crackhead burglar bustin' down your door at 2 am are really, really remote, so remote that most Canadians don't feel that they need a gun to defend themselves. Despite that, there is one registered gun for every three Canadians, that's quite a high rate of ownership; you don't hear about it though; we tend to not point guns at each other as much. Gun related crime is not only lower that the US, but France and Germany as well.
Sure there is a distinct Canadian culture, we believe in "we" much more than "me". We believe that society as a whole bears some responsibility for human dignity. We accept diveristy and differences as a strength. even celebrate it. Which is all part of why we live longer, test smarter, rank higher as a place to live and are even generally wealthier than our neighbours to the South.