In the linked song below, are they pronouncing 산토끼 correctly?

ANSWER: 산토끼 literally means "a mountain (산 山) rabbit (토끼)," which refers to a wild rabbit, so the 산 in 산토끼 should be pronounced with a short vowel sound since the 산 for "mountain" is pronounced with a short vowel sound. If you pronounce it with a long vowel sound /사안토끼/, then is means "a live rabbit" (산 토끼) since 살다, the word for "live" or "alive," is pronounced with a long vowel sound.

By the way, the word 산새 literally means "mountain (산) birds (새)," so the 산 in 산새 should also be pronounced with a short vowel sound. In addition, the 새 in 산새 should be pronounced as /쌔/. If you say /사안새/, it means "a live bird" (산 새), and if you just say /산새/, it means "a bought bird" (산 새). So, /산쌔/ means "a mountain bird"; /사안새/ means "a live bird"; and /산새/ means "a bought bird."
Finally, what does 새머리 mean?

ANSWER: My dictionary says it means "spareribs" and is pronounced as /새애머리/. When the 새 for "bird" is the first syllable in a word, it is pronounced with a long vowel sound. This makes me wonder how Koreans say, "bird head." Because even if they say, "새의 머리," it would probably still sound a lot like the Korean word for "spareribs" since most people are probably too lazy to pronounce 의 as /으이/ and just pronounce it as /에/.