Will connecting Vcc and enable together cause problems
The device you have chosen isn't meeting your expectations because it doesn't work very well driving up to 1A motors. Here's a the circuit: -



The output stage is really old hat and poor for driving an amp or more on low supplies. The upper transistor in the output stage will barely be able to get to within 0.7V of the supply and the lower transistor will probably just about get to 1.4V of ground. This is the main problem.

If you look at page 4 it tells you this: -


Low level output voltage can be as high as 1.8V (not 0V) for a 1A
load
High level output voltage can be down at Vcc - 1.8V for a 1A load


As for the inputs floating up this is to be expected given the input circuits used within the device.

Regards the circuit diagram in the question - I believe the external diodes are added because it is driving a two phase motor and this might create extra problems that the internal diodes can't cope with.

Also be aware that the continuous output current for this device has an absolute max value of +/-1.1A so be aware of that when choosing motors. There are far better devices than this old dinosaur by the way.