I'm not following how the bubbles confirm any leak...the bubbles kind of look like they are produced from decay gasses and soil settlement to me.

The contractor said he dug the core "deep", but did he dig down to and tie it into good clay? How was the core compacted? Was any of the rest of the pond clay lined and compacted?

A pond can be built over springs if a suitably thick, well compacted clay liner is created. If springs flow into your pond, they can also become drains as water seeks it's lowest level..