Translation:Please stick a stamp on the postcard.
葉 is not used in this word to literally mean "leaf". It is 当て字 (ateji), a kana used for its sound rather than its literal meaning. The kanji phrase 葉書 is not outdated or unused, rather the characters are time-consuming to write, and since the object it refers to is a common household object, most people write it in kana instead to save time. You will still see 葉書 used in formal writing.
I think what HibaAkaiko meant was that katakana is used as an alternative to kanji, in cases where you want to make a noun stand out from the rest of the sentence.
In an all-hiragana text, Japanese's lack of spacing might make it difficult to tell where one word ends and another begins. To make the distinction easier, but without wanting to add complex characters, they often write it in katakana instead. For the names of animals it's almost always done like this, and it can also be a way of putting emphasis on a word (the same way we'd use italics, for example).
I dont see how thats accurate. Most words that have kanji are written in hiragana when they arent written in kanji, so im not sure if youre trying to say something different or misinformed.
Words that are written in katakana are almost always loan words or foreign names. Unfortunately i do not know where "hagaki" comes from, hopefully someone else knows.
What he meant to say was that words which have complicated kanji are often written in katakana to make it easier to write, especially by children. In formal texts, however, these words are always written in kanji. There are three reasons why a word may be written in katakana: (1) it is a loan word, (2) for emphasis, or (3) the kanji is too difficult.