They are different sentences. Word order matters. This sentence specifically says "There are birds in the forest." To say "the birds are in the forest", you'd have to say "птицы в лесу". This sentence is about what stuff is in the forest, not about the birds themselves.
When do we use the locative/prepositional? Do all nouns have a locative form? Any rules/hints?
I believe the notes should have something on the Locative (also known as the 2nd Prepositional). Basically, you always use it with certain nouns (verbs have nothing to do with it) when talking about physical or temporal location with the prepositions «в» and «на». So, for example, «быть в лесу́», but «говорить о лесе», «лежать в снегу́», but «думать о снеге». There is a set of about 150 nouns that use the Locative, some can be found here.
Is the speaker saying птица or птицы? It sounds to me like the former, especially in slo-mo.
What's wrong with "There are birds in a forest"? One wouldn't say that very often, but grammatically speaking, wouldn't this suffice?
A poem commenting on the meaning of life and it's relation to social media: There are birds in a forest/ And people in a city/ All they do/ Is tweet, tweet, tweet