An umbrella stroller is a light-weight, folding baby stroller that usually features upside-down-J-shaped handles similar to the handle on an umbrella.
The Graco stroller does not count as an umbrella stroller. I just checked NWAs policy and they only allow a fully collapsible umbrella stroller as a free item.
http://www.nwa.com/travel/luggage/carryon.html#personal
As long as you gate check your stroller, it will not count against your checked baggage limit (unless you are flying international). If you check the stroller with the rest of your luggage, you run the chance of them counting it towards your checked baggage limit since it will take up space in the hold.
Gate checking means that you are allowed to use your stroller up until you board, and it will be waiting for you as soon as you deboard the plane. Here is how gate checking a stroller works:
When you first check-in, let the agent know you have a stroller. They will put a a luggage tag on your stroller just like they do the rest of your luggage, except you are able to use the stroller throughout the airport. They will also give you the claim ticket for your stroller like they do the rest of your luggage. If you plan on using a stroller bag, le them know and they will give you the tag to put on the bag later once you have your stroller securely inside it, but before boarding.
When you go through security, you will be asked to fold your stroller and place it on the conveyor belt to be X-rayed. Because of this, it is always best to make sure that any "extras" you have on your stroller are taken off so that they are not damaged and so that the stroller can pass easily through the X-ray machine.
When you are boarding, whether you board first, in the middle, or last, there will be an agent that will take your stroller from you. Again, make sure that any extras are taken off so they do not get damaged during the flight, and secure your stroller in the closed position. The agent will place the stroller in a special compartment under the plane for the duration of the flight.
When you exit your flight, the handlers will have your stroller waiting for you at the end of the jetway, near the gate. Should you have a bus arrival instead of a gate arrival, they will place the strollers on the ground near the buses for parents to collect.
I've been flying internationally & domestically with my children since they were each 2 months (now 8&6, 9 months and currently pregnant with #4). At least three trips a year are the children & I traveling internationally (14+ hours each way) alone. We do another 8-10 international & domestic flights a year as a family. Due to this, I have written a small article about flying with children that goes more in-depth, gives up to date security rules & regulations for the US, EU, UK, & Australia, offers tips & tricks for the travel, etc.
http://www.angelfire.com/jamiehassen79/flying_children.html
Hope you have a great flight, an if I can be of any more help or assistance, please feel free to contact me.