I've been to Disney World twice, both times staying at a value resort (all star music and last trip pop century) and requesting a king bed in advance.
Both times the first night was given a room with two doubles (we had to request a room change both times) and the second night on both times we got what they call a king bed (it's just the two doubles pushed together, why they could have done that is advance for our first night I've never figured out). I've been told by different cast members and travel agents that to get a true king bed at a Disney hotel you have to book at a deluxe or above hotel
The king bed room are indeed the handicap acessible rooms (you won't get a tub in the bathroom just a walk in shower (make sure you don't leave any clothes or towels on the floor when you shower, as the bathroom floors get wet due to the low lip of the shower area)).
Strangely both times we were moved to rooms on upper floors so upper level accessible rooms do exist.
The cool thing about changing rooms is they will move all your bags for you during the day, so you don't have to wait or carry all your stuff. Just call the number they give you at the check in desk tell your name, that you have a room change request, your current room number and the number of bags go to the park for the day and when you come back go to the check in desk and they'll give you a new key card and a new room number.