Translation:I put the postcard in the mailbox.
Possibly to indicate that this is a noun and not part of the sentence (i.e. to avoid confusion with the particle には).
They use all three ways of writing it, but it IS usually in hiragana. The thing with the katakana is that they are sometimes used for common hiragana words. They get written using katakana for extra emphasis and/or easier identification, so you are probably right. In some sentences that might be a problem.
Funny enough, they also use ポストカード too? I get very similar image results when I search for each term, though ポストカード seems to show more souvenir or graphic cards, and perhaps ハガキ might be more of the everyday type? I'm not sure about that.
ハガキ is actually one of those random words that for some reason is written using katakana quite often, despite it not being a "foreign word". It's very probable this is done to improve readability, since it's so easy to mistake a は in the middle of a sentence as the topic marking particle rather than the first syllable of a noun.
An easy way to tell which pronunciation is correct in the sentence is to check whether the sentence has a direct object or not. はいる is intransitive; いれる is transitive.
The first time I heard this was when someone said だが and it sounded like a da-nga sound. It happens and, as it seems, it is natural.
That's how it can sound... You'll get used to it yet. Knowing the word you misheard it like doesn't work in the context will help out a lot. (A lot of words in Japanese do in fact sound the same, so it's a common skill among the speakers ;)
Japanese doesn't differentiate between definite and indefinite when there is no pre-noun adjectival (その, この and the like), so lacking further context both should be accepted.