In conventional speech people sometimes use words which are gramatically incorrect. Хочем is gramatically incorrect, but this singer doesn't care of any rules :) I wouldn't say that the song is sarcastic, they just say that they want to go to Sochi and they don't care of London or Paris.

"Хочем" is a dialect variant. Usually people originally from Ukraine or from the south of Russia can use this word in their coversation. But you shouldn't use it, it's not correct usage of this word in Russian. In the video the singer also repeatedly said "шо" instead of "что". It's also a dialect variant (Ukrainian or in the south of Russia people say like this). "С дочей" - Ukranian variant or south of Russia again. In Russian the usual correct variant is "с дочкой".

It is definitely very tongue-in-cheek, you are absolutely right. )

It is Russian mockney.

This is imitation "Uneducated adult Russian". "шо", "чо" correct "что", "хочем" correct "хотим". "С дочей" - more dearly than "С дочкой". Yes, the song is sarcastic for singer, but it is genuinely for "Uneducated adult Russian".