"There are three birds."
Yes, "３匹の鳥がいます。" (with the の as mentioned) is correct for this sentence, but it is more diverse in its usage which can affect the meaning of the sentence. When adding the counter after the particle ("鳥が３匹います。") it can only be used to specify an amount. This makes it less ambiguous when the number does not necessarily specify an amount.
Here is an example where the location of the counter does make a difference: 三階の建物に行きます。 I will go to the 3 storey building. 建物に三階行きます。 I will go to 3 floors of the building.
So it is safer to place the counter after the particle if you want to specify an amount.
You are correct! As a stand alone, that is the kanji for feathers, wings, and blades on things like helicopters. As the suffix in a compound kanji, is also a counter for birds and rabbits.
「一羽 」いちわ "one bird";
「六羽」 ろくぱ "six birds"; 「 白羽」 しらは "white feather";
「白い羽」しろいはね "white feathers"; 「羽」はね a (bird) wing [insect wings are usually the kanji 跳, although still pronounced はね].
It can also be combined with other kanji to make things like:「 羽化する」うかする "emergence [of insects]" (like from a chrysalis; and「羽毛」うも "plumage, down." (I think it also looks like the kanji for ”weak” 「弱い」よわい.) Hope that helps!