Is the rotation speed of the Earth slowing down?

Yes it is. Astronomers can measure this very precisely now that they have atomic clocks to provide a non-celestial time standard for comparison. The length of the day is increasing by 0.0015 seconds every century, of which about 0.0007 seconds per century has to do with the tidal breaking of the Moon. As a result of this, the Moon's orbit must also increase so that the Moon is slowly getting farther and farther from the Earth by a few centimeters per year or so ( my estimate!). As this process continues, it is predicted that in billions of years the lunar month will increase to about 47 days from its current 27.3 days. But by that time, the Sun itself will have begun to evolve into a red giant, which will upset the Earth-Moon system somewhat; especially if they are both engulfed by the Sun's expanding atmosphere!