To answer the first part theoretically, yes. Practically more often no than yes. Good examples for the 'no' camp are such as the Amazon tablets and Fire TV systems and the AOSP devices which ship without Google Play. These are more common in the Chinese and Indian markets (though grey-to-black market phones/tablets/TV sticks/TV boxes can be found), because shipping an Android device without Google Play in Western markets is not going to fly at all, to the point where specifically advertising Google Play is positive marketing.
You can try going Play free yourself. Flash a clean ROM from places like XDA, skip the Google Play packages and you can get by with 3rd party markets like the Amazon Appstore, Getjar or F-Droid. I definitely wouldn't say most apps require Play Services, just from looking at what is on offer in other markets. The ones that do may do so for verification, Google specific integration or some bells and whistles that Play Services provide.
But to your last point, most Android devices ship with a standard file viewer which will not contain ads. The ones you mention are 3rd party, made by someone who did not get paid when you bought your device or use Google's services. For their time and effort creating the software, they are entitled to making some money.
The explorer has been part of Windows for ages, and most would consider that having paid for Windows, Explorer was a component which was also paid for.