External Hard Drive/File Transfers

First of all sorry to be that guy who has some potentially easy questions.

I'd like to start by saying I consider myself pretty technically savvy. However I haven't done much work in the way of Apple products.

My mom just bought a new Macbook Pro and is giving me her Macbook Air, but first I'm in charge of transferring all of her data over to her new computer. I attempted a wireless transfer with the help of migration assistant and after it failed for the fifth time I gave up on that. Turns out she has a 500GB WD My Passport that she has used once when she was having internal hard drive problems. Now I plugged it into the Air, my question is how do I know that it's working? It is formatted and has been used once before, but I didn't see anything pop up beside the hard drive icon on the desktop. I opened the hard drive in the finder and theres are two arrows moving in circles next to the icon which I'm assuming means it's backing up, but I just am not 100% sure.

My next question is how long should this take if it is already backing up? It's been several months since the last backup.

My last question is can the My Passport be used to move all of my mom's files from the Macbook Air to the Pro? If so is it as simple as plugging it in and copying it over.

Thank you very much.

If it's been several months since her last backup, then there is probably a huge backlog of backups waiting to go to that drive. Ever since Lion (10.7), the MacBooks will automatically keep a temporary Time Machine backup stored on the internal drive, and will copy the backups over when the designated backup drive is connected. Since this has been so long, I imagine it may take a few hours.

Once done, take that to her new MacBook; plug it in; and run the Migration Assistant. You will have the option to migrate a user off of a "Time Machine backup", which would be that drive.