Learn Russian: На пляже есть свободный шезлонг?

Свободный means 'free' in the sense of 'unoccupied' or 'available' (like an empty chair or having free time). Бесплатный means 'free of charge' (costing zero money). Here, you are asking if an empty deckchair is available to sit in, not necessarily if you can use it without paying.

Since our current lesson is about moving into a position, why is it на пляже and not на пляж?

На пляже is in the Prepositional case because it describes a static location, answering the question 'where?' (где?). You use the Accusative case (на пляж) when talking about directing an action toward a destination, such as walking to the beach or lying down onto the sand.

If I find a free deckchair, how would I say 'I am lying down on the deckchair' using the verbs from this lesson?

You would say Я ложусь на шезлонг. Because you are actively assuming a position (a change of state), you use the verb ложиться (or the perfective лечь) followed by на plus the Accusative case (на шезлонг). Once you are already settled and resting there, it becomes a static location: Я лежу на шезлонге (Prepositional case).

Why do we need the word есть here? Can we drop it?

When asking about the general existence or availability of something ('Is there a...?' or 'Are there any...?'), you need to use есть. If you drop it, the question shifts focus to the location of a specific known item, sounding more like 'Is the free deckchair on the beach (as opposed to by the pool)?'