Can you exit the airport when transiting the Schengen area on a single-entry Schengen visa?

Yes, you can do that as the number of entries refers to the whole Schengen area, not to Spain specifically. You will therefore enter the Schengen area in Amsterdam, whether you go sight-seeing or not, and travelling between the Netherlands and Spain does not count as an exit/entry.

Because of that, it's important to have a visa valid for the Netherlands and not only for Spain. It's usually the case for most Schengen short-stay visa but you should check what it says under “Valid for”/“Valido para”. If it says something else than “Estados Schengen”, you might have a problem.

Yes, your passport will be checked when you exit the non-Schengen area of the airport. This check is done before you can choose to exit the airport or to take a Schengen flight.

So you enter to Schengen area only once. When you go back to Amsterdam airport, you will not pass any passport control.

Note: because of security measures on airports, you may show your passport at some checkpoints, at x-ray, and/or at gate, but these are only identity control.

The Schengen Area is a bloc of several dozen European countries in which travel and visa handling are handled at a supranational level. In short, that means that a Schengen visa applies to the bloc as a whole, and not to individual countries.

Once you have permission to enter the Schengen Area, you are free to travel within that area, and you will not encounter any border controls, although you may be asked to produce the necessary papers by officials during 'non-permanent checks', such as highway patrols.

To answer your question directly: you will be allowed to leave Amsterdam airport to see the city (taking a direct NS train is cheap and convenient) and come back for the second leg of your journey. Do note that though you will not have to pass border control and customs, you will have to pass through airport security again, which can take some time depending on the time of day and season.

Enjoy your trip!

// Edit: You will be entering the Schengen Area in Amsterdam, much like one enters the US in Newark of JFK, even though there may be another leg to another US city such as Houston. For visa purposes, the leg from Amsterdam to Barcelona is domestic, just as JFK to Houston would be.