Is bluetooth 4.0 backwards compatible with bluetooth 2.0+EDR?

They are backward compatible. But - since each version brings new functionalities with it, only the basic function set is backward compatible all the way, see the table for more detail. In addition to that 2 devices wanting to communicate using bluetooth have to support the same profile(s), the full piece that i copy/pasted from here.

Not if it is "Smart"

In general, Bluetooth is backwards compatible: Bluetooth devices supporting the just-announced Bluetooth 4.2 standard should still be able to pair with devices using, say, the ancient Bluetooth 2.1, launched back in 2007.

The exception are gadgets or peripherals based on the low-energy version called Bluetooth Smart. The protocol of Bluetooth Smart differs from that of older, so-called "Classic" Bluetooth devices. Bluetooth Smart devices are not backwards compatible and will not recognise nor pair with older devices supporting Classic Bluetooth. For example, an old Sony Ericsson phone sporting Bluetooth 3.0 will not be able to connect to a Bluetooth Smart device.

However, if a device supports Bluetooth 4.0, it can potentially recognise both Bluetooth Smart and Classic. If it does, it is officially labelled Bluetooth Smart Ready.

Personal health gadgets such as fitness bands and heart-rate monitors commonly employ Bluetooth Smart. These gadgets will only pair with smartphones or tablets that are Bluetooth Smart or Bluetooth Smart Ready.

[Source]