Can I have too high a PSU power rating?

There shouldn't be.

The wattage is the measure of how much power your PSU can deliver. The components will (unless they are faulty) only draw as much as they need.

Having a too low an output PSU will cause the PSU to fail in the first instance, which may cause other components to fail as a side effect.

In relation to your edit - there's not really a "compatibility" issue with power supplies. You can get converters (molex <> sata) and splitters if you need to connect older devices. The main thing to ensure is that the combined power consumption of all your devices (hard drives, motherboard, graphics card etc.) is less than the output of the PSU. With a 650W PSU should have a good buffer.

The "W" in 650W refers to the wattage not the voltage. The higher wattage will not cause any problems for your PC

There's no problem, you can have a PSU way more powerful than needed. In some case it's even recommended.