Setting Up Roaming With Multiple Airport Extremes

This may help: Wi-Fi base stations- Setting up & configuring a roaming network (802.11 a/b/g/n).

I am thinking of purchasing two Airport Extremes and setting up the roaming feature. If I were to walk downstairs with my laptop while it is connected to the upstairs router, would it have to disconnect from the internet to connect to the downstairs router, or would it do it seamlessly like the Ubiquit Unifi systems?

In theory, and in general practice, the laptop will "roam" between both base stations so that you will not need to disconnect it from the Internet to do so.

Can I also use airprint on a printer that is plugged into the upstairs router while using the downstairs one?

If the printer in question is AirPrint-capable, then yes, you can print to it regardless of which base station it is connected to. Note; however, connecting a non-AirPrint-capable printer into the base station's USB port will NOT make it AirPrint-capable. Most AirPrint-capable printers have built-in wireless so they would connect to your wireless network and, not necessarily to a base station's USB port.

If you have a Mac laptop, then it will usually automatically switch from one Apple access point to another as it moves from one area to another. But, other devices like the iPhone and iPad will not do this very well, if they do it at all.

With the iOS devices, you will need to get in the habit of temporarily turning the WiFi off on the iOS device, then move it to the new location and turn the WiFi back on, and it usually pick up the closest Apple access point.

An AirPrint capable printer is almost always going to connect to the wireless network, so not sure what you mean when you say "plug in". Would this be an Ethernet connection? A USB connection will not allow AirPrint to work correctly.