The reason why your rabbit is lying down is because the rabbit is tired, are resting, or they are feeling safe and content.
Rabbits are naturally crepuscular and will also nap during the day as well so when they lay down they could be sleeping or resting.
In some cases rabbits lay down when they are relaxed and is a strong indication that the rabbit feels safe and content.
The reason why rabbits lick people is because they are showing affection and trust.
Rabbits licking you is similar in how they groom each other as well and they strengthen the bond they're sharing.
Rabbits also lick people when they're investigating your skin, clothing, scent or are tasting it.
Rabbits stare at you as a sign of affection.
Rabbits also stare at you out of interest and curiosity.
Rabbits are naturally observant animals and often assess you as a potential companion or they sometimes simply are seeking out your attention.
When a rabbit stares at you it can also mean that the rabbit is looking for a treat or they're hungry depending on the rabbits body language and context.
In most cases when a rabbit is staring directly at you and they haven't eaten yet it can mean that the rabbit is hungry and they know it's time for you to feed them.
But if they've already eaten the staring from the rabbit is often just a sign of affection or staring for attention.
Rabbits are also prey animals and so they are naturally watchful and often will stare at things to assess any potential threats or changes in their environment.
When a rabbit is staring at you when they are relaxed it is often a positive sign and they are often wanting to be petted or be interacted with.
And if the rabbit is staring at you while also hopping around or also nudging you, then they could be asking for treats or seeking attention.
Rabbits also use their gaze in order to assess if you're a potential threat or a safe presence for them.