the voice over clearly says bachadar not bechadar - therefore I sleep in the bedroom should actually be the correct answer as opposed to i sleep in a bedroom
Well observed. But no. "In the bedroom" has to be בחדר השינה, this is a construct and so the second word takes the definite article. The reason this is pronounced "bachadar", and that only in formal speech and not in daily conversation, is one of those obscure rules of Hebrew nikkud change.
I am a little unsure what you teach. Do you say that to say "bakhadar shena" would be better than to say "bekhadar shena" . Sometimes it can be a lot of contractions of sounds between words in the audio, so like in this phrase I thought the hey before shana was consumed and I trusted the "ba" sound to make the word definite. Is the "be" sound for indefinite only used in formal hebrew?
Yes, בַּחֲדַר־ is of high formality and is therefore "better" in a puristic sense. There is nowhere a He, consumed or not consumed and the word is indefinite. The a-sound of [ba] is only triggered by the very short, sheva-like sound after the ח, to which the vowel of the preposition assimilates (If an inseperable preposition is joined to a guttural letter having a compound sheva, the prefix takes the simple form of the vowel). In other words, it is the other way round, the [ba] is only used in special cases for the indefinite in (very) formal Hebrew.