Learn Dutch: Bovendien liggen er daar geen luchtbedden. - Moreover, there are no air mattresses lying there.

This is a perfect example of avoiding a double er. The sentence needs a dummy subject er because the real subject (geen luchtbedden) is indefinite. It also needs a word for the location ("there"). Normally, an unstressed "there" is also translated as er. However, Dutch strictly avoids saying er er. To solve this, the locational er is substituted with daar. You could also just use a single er (deleting the locational one entirely), but adding daar puts a slight emphasis on the specific location.

Could I use zijn instead of liggen to say "there are no air mattresses"?

You technically could, but Dutch strongly prefers positional verbs (staan, liggen, zitten) to describe where physical objects are located. Because air mattresses are flat and lie horizontally on the ground, liggen is the most natural and descriptive choice here. Using zijn would sound noticeably foreign to a native speaker.

Why is the verb liggen placed before the dummy subject er?

This is due to the V2 (verb-second) rule. The sentence starts with the adverb bovendien (moreover), which takes up the first grammatical position. The conjugated verb liggen must come second, causing inversion and pushing the subject er to the third position.