Have you moved the margins (Magic Margins) to their extreme left and right positions? Does the space bar move the carriage forward? Did you try using the margin release key? Based on your description it sounds like your margins have been set next to each other effectively locking the carriage and the typebars in one position.

As for the model name, it's just HH. The E in the serial number only designates that it has been fitted with an elite typeface.

Looks like you have a 1952 model (judging by the "Royal" insignia being cast in the entire front plate rather than as an inlay into the front plate. The ribbon color selector is also straight across, versus the zigzag pattern you'll find on later Royal HH machines dating to 1956.  Look underneath the keyboard and you'll find a long plate right next to the comblike key guide.  This thing pivots backward to the key guide to prevent the keys from being pressed when the right margin is reached.  Sometimes when the carriage is returned to the left margin, this plate doesn't go back like it should--it sticks there either because it needs to be cleaned and lubed in its pivot or the whole mechanism is needing cleaning and lubrication.  Also, exercise the mechanism once cleaned and oiled, and it should do okay after that.

I didn't say these sere exclusive to the '52 models.  I have seen many times something occurring in more than one year, with changes occurring in the middle.  But, while I have you, can you tell me if they made any HH machines in 1957 before they started with the FP?  Or did the FP start in mid-'56?