Ма́льчик is a 'boy', 'child' is ребёнок (not sure how it's written in the transliteration used here; rebyonok?).

No, it's the other way around - "hen" is "курица", "петух" is "rooster".

Hen is a bird, chicken is a food or chick. When we say in cafe, for example "Я хочу курицу", it translated as "I want Chicken"

And then there's цыпленок which is apparently a chicken of some sort.

Indeed, ку́рица 'chicken, hen' sounds similar to ку́рится 'is smoked', a form of кури́ть 'to smoke'.

In careful speech, ку́рится would be pronounced with long ц, like ку́рицца, while ку́рица is pronounced with short ц. But this distinction is not always observed, especially in fast speech.

There are puns based on this similarity, for example:

— Официа́нт, это ку́рица? 'Waiter, is this chiken?' [pronounced same as это ку́рится? 'is this [to be] smoked?']
— Нет, э́то ку́шается. 'No, this is [to be] eaten.'

Both the audio here and the Forvo recording sound the same to me. It is pronounced /vʲidʲɪt/, with the palatised versions of the consonants /vʲ/ and /dʲ/.