Bedroom is more complicated, and I think it is more a personal thing than a set rule. I would say 'Be droom' - to me the 'd' and 'r' are closely pronounced in a sort of affricated way, and I don't say the word like bed + room.
Generally, in syllabification of English, you want to have a consonant starting a syllable. So, in a word like 'walking' it would be 'wa(l) king', not 'walk ing'. I put the 'l' in brackets because it isn't pronounced.
Also, remember that syllables aren't really describable by their nature. It's an abstract concept that humans can measure with some ease, but instrumental analysis can't. Linguists don't really agree what makes a syllable a syllable - just that vowels are important.
Apart from helping children/adults with word finding difficulties, writing poetry/lyrics, and analysing speech sounds, I never really see any practical use for splitting words into syllables.
I know, for example, in Portuguese, syllables are counted considerably differently than in English.