f# and DX/FX questions

1. Is an ƒ 2.8 DX lens as bright as an ƒ 2.8 full frame lens, i.e. is there any loss of low light performance beyond that expected by the smaller sensor size?

2. Will a lens like the AF-S DX 35mm f/1.8G have deeper depth of field at the same aperture on a DX camera than a 35mm f/1.8 made for full format sensors when used on a full frame camera?

3. I'm trying to understand if the aperture number on a lens like the AF-S DX 35mm f/1.8G means exactly what it says, or weather it needs correcting to a 35mm equivalent like the focal length (which is actually 35*1.5)?

Thanks!