Transistors are closing slower than opening - will it be problem in H-bridge?
Is that right conclusion? Is that a real problem? And if it can be
  corrected/used, what is good PWM speed to drive it effectively?


Yes this can be a real problem and one solution is this: -



The RC keeps OUT1 and OUT2 from being on at the same time. Make RC equal to a few microseconds to avoid the "clash". See also this stack exchange Q and A.