Question 1: You can capture a static snapshot of the basemap to your local drive, and use that. This involves turning off all layers except the basemap in dataview, zooming to your area of interest, then choose "Export Map" and select to PNG or TIff, and select "Create World File." You will get a georeferenced image on your local machine, that you can use independant of the image / basemap server.
Question 2: Once you have selected rectify image, the georeferencing is built into the file. However, it is in the new file created, not the original, so add the newly created file to your map and use that instead of the original. To update the original file, choose "update georeferencing". Then you can move on to the next aerial and the first should show in the position where you have relocated/georeferenced it.
Regards,
Jim