I've found various references but no clear explanation so forgive me if this is a topic previous visited.

Are 2/3 lens mounted on the EX3 fully functional?

Most 2/3" broadcast lenses will be functional with the right adapter, however, due to the smaller 1/2" sensor your field of view will be about 35% narrower. In other words, an 8mm lens on the EX3 would feel like an 11mm lens would feel on a 2/3" camera. Most people (including me) wouldn't like using that as their normal lens. So, you really need to have a very wide 2/3" lens, unless you only care about the telephoto end of the spectrum.

Here's the adapter you'll need.
Fujinon ACM-21 2/3" Lens Adapter for Sony PMW-EX3 ACM-21

If you haven't bought a 2/3" lens yet, you'd be a lot better off buying a 1/2" lens for the EX3. I used my Fujinon HSs18x5.5 lens on my EX3 100% of the time and loved it. Changed the "feel" of the whole camera.

The MTF EX3 to 2/3 B4 adapter is a lot cheaper than the Fujinon adapter. Powered B4 to EX3 Adaptor : Lens Adaptor.Com (http://www.lensadaptor.com/lens-adaptors/b4-adaptor/powered-b4-to-ex3-adaptor/)

Using this adaptor with the 2/3" lens, finally the DOF on a EX3 will be like a 2/3" sensor or 1/2" camera sensor?

DoF is governed by the sensor size, so DoF will be 1/2".