What are constructs? Relics, resources or what? They don't have an actual relic roll and they cant exist on the map without the battery or a character with a construct. If you have a resource, can you play a character and its construct?

A construct is niether a relic nor a resource. They are "constructs" which is a thing of it's own.

So yes, you can play a character with a construct and still use a resource.

They're just like the word balloons. A specific game element that can only be played under specific circumstances.

Constructs, Word Balloons, and any future similar thing that attaches to specific figure's bases will not interact/conflict with any Relic/Resource unless one of them comes with specific rules saying so.