Can an essentials character, according to the rules, take core edition powers?

When applicable, yes. Sometimes they're very limited, such as a Thief can take Core Utility powers, but could not replace one of their "Tricks" with a regular Rogue At-Will.

But many of the entries, such as the Sentinel, say something to the extent of "Choose an at-will. Here are five, in case you own no other books."

The character builder is usually my go to, if it's terribly unclear what's allowed. If you click the "Show additional Options box", it'll let you pick from the rest of the class' powers, if it's permitted.

Ditto. In most cases there is nothing stopping you from pulling from either source.

The martial classes are generally out of luck since the Essentials versions function so differently from core. But most other classes can easily draw on their standard spell list.