In order to decide which verb tense to use, you often need some context: what have you seen, where, when, and what do you know about this situation? If you see a dog lying at its master's feet, you can easily say "The dog is lying at its master's feet (now)". But if you see it every day, you can also say "The dog lies at its master's feet (every day)". Or, if you saw it lie down half an hour ago, you can also say "The dog lay at its master's feet (half an hour ago)". Or maybe you remember seeing it yesterday: "The dog lay at its master's feet (yesterday). (I don't know what it's doing now)."
Or maybe, importantly for your question, you have just seen the dog lie down, so you could say "The dog has (just) lain (down) at its master's feet". By itself, it's an awkward sentence. Even in context, it's still awkward. Youtube presenters don't always choose the best example sentences.