As far as why the existing laundry room 120V receptacle does not have GFCI protection, it is quite likely that it was not required at the time. GFCI has, generally, been required longest in kitchens and bathrooms with laundry rooms and other areas added later. In addition, many jurisdictions are consistently several years behind the latest NEC code. A quick search says the requirement for receptacles nearly laundry sinks was added in 2005.
You can certainly add GFCI to the existing circuit, either at the breaker panel or, generally at lower cost, by replacing the receptacle with a GFCI/receptacle. But you don't have to do that to existing receptacles.
On the other hand, if you add a new 120V receptacle in the laundry room, it should be GFCI protected. (240V is an interesting situation, as the latest code may require that as well, with "latest" depending on your jurisdiction.) If you are extending it off of a bedroom circuit, just use a GFCI/receptacle in the new location and you're all set.
There is nothing wrong with multiple 120V receptacle circuits in the laundry room - I have at least 3 (dedicated washing machine circuit, telephone equipment (used to be shared, separated with my panel replacement), convenience receptacles (shared with another room)). The key code requirement is that you have one that is dedicated, normally used for the washing machine. Note that this is quite different from kitchens and bathrooms. Kitchens and bathrooms have requirements (two circuits for kitchen, one for bathroom) for dedicated circuits that can't be shared and additional circuits are, as I understand it, also not supposed to be shared with other rooms. ("Other" is a bit fuzzy - kitchen includes certain dining room and other areas. For bathrooms, multiple bathrooms can share a circuit.)
Receptacle height should not be an issue. The only problem, aside from ADA requirements, which are more of an issue in a commercial setting, is that you can't have a GFCI/receptacle (GFCI protected is OK, the issue is "getting to the TEST/RESET buttons") too high up or blocked by an appliance. Waist high is actually ideal - easily accessible by anyone except the smallest children (who shouldn't be using the receptacles anyway), and in fact is closer to the typical height of kitchen countertop receptacles. The receptacles in my laundry room are all around 3-1/2 or 4 feet up.