Is there an origin difference or English reason why "living room" is two words, but "bedroom" is only one?
"Living room" has been in the language a while; the OED's first quote for it is from 1787. I think it hasn't become a compound word due to "living" ending in "ing". Words that end in "ing" seem to only very rarely form the first part of closed compound words.
Looking at the OED, there are over a hundred entries listed that end in "ing room" or "ing-room" but none that end in "ingroom". (If you have access to the OED, you can see this by searching for
*ingroom.) The same thing can be seen if you search for
living*, to find all the entries that start with "living". There are very few entries that were a single word (none of which were compound words, as far as I saw). Most entries had a space and one had a hyphen ("living-in", which as an adjective is different).