Both depth-of-field and and total image noise are closely proportional to the total amount of light reaching the sensor, so thinking of the lens as the same as a 40 f/3.4 in a FF camera is a good analogy for all practical purposes in terms of the final images you can get.
No. Depth of Field and background blur are much more complicated topics than that.
The image circles produced by a 20mm f/1.7 lens for points outside the focus plane are the same size as the corresponding circles produced by a 40mm f/3.4 lens; this is the basis for the claim that a 20/1.7 is "equivalent" to a 40/3.4. However,
a photo taken with a 4/3" camera will be enlarged twice as much as a photo from a full frame camera . This means that for a proper DoF calculation, you also have to adjust the acceptable circle of confusion criterion to make it more stringent for the 4/3" output—twice as stringent. This means that on the final print, the 20mm f/1.7 on a 4/3" camera will generally have
less DoF than a 40mm f/3.4 on a full frame camera, and will in many situations exhibit just as little DoF as a 40mm f/1.7.
The idea that a 20mm f/1.7 lens on 4/3" has a lot more depth of field than a 40mm f/1.7 on full frame comes from two mistakes:
- Confusing depth of field with background blur. In cases where the two lenses show the same DoF, the out of focus areas from the 40/1.7 will be a lot blurrier. People assume blurrier = less DoF, but DoF isn't how blurry the unfocused parts of scene are; DoF is about how much of the scene is acceptably sharp.
- There will be many circumstances where the 20/1.7 on 4/3" will indeed show more DoF than the 40/1.7 (but less than the 40/3.4), especially at longer focus distances and with smaller prints. This is because when the degree of blur falls under the CoC criterion, suddenly things "snap" into acceptable sharpness, and this happens more easily with the 20/1.7 and its smaller blur.
Check out this page, especially figures 3-5 and the sections of the text they appear in:
http://toothwalker.org/optics/dof.html