Drywall – How to replace ceiling drywall over cabinets

The 2X4's you are looking at are collar ties attached to the end of the rafters and span wall to wall. They are really only strong enough to support ceiling materials, not a live load. The new floor joists will need to rest on the upper wall plate (load bearing) adjacent to a rafter, and cross supported.

The items you need to determine are span, joist spacing and material to be used as floor joists. This link shows a chart in feet and inches for you application: http://www.the-house-plans-guide.com/joist-span-tables.html. Even on short spans, I would never go any smaller than 2X8's to assure a flex free floor.

The second issue you will face is insulating the roof. The size of your rafters are going to dictate how much insulation you can install. If they are less that say, 2X10, you will not be able to get a good R-value. You may have to consider maxing out the cavities with blanket insulation, then also adding some 2 inch rigid foam sheets over the entire field to increase the total R value. You could then install strapping (purlins), with long screws, to mount drywall to. In this kind of roof design, there would be no venting required. This is a very common type of construction on cathedral type ceilings. Good luck.

The overkill direction (which is actually appropriate for some circumstances, such as a ceiling that needs to hold up a lot of insulation, as for an attic above) is to run 1x3 strapping opposite to the joists. An advantage in the application I mention is that this strapping can be on 12 inch spacing, so the drywall can be attached with a screw every foot, even if the roof trusses are on 2 foot centers.

In general, it is actually advantageous to leave the edges of the ceiling to float for the last foot or so, so long as you drywall the ceiling first and the walls after, butting them up to the ceiling before screwing them in place - the wall drywall then ends up supporting the edges of the ceiling drywall, but the whole is slightly less prone to crack in the case of small building movements as there's a little give from the floating edge. Here's a reference (they also suggest using corner clips.) Scoot down to photograph 4, and you won't be surprised to find that they mention the other point of view as being commonly held. As for it being "a developing point of view" only in building would something code approved for 30 or more years (per the above reference) be "developing..." An additional reference. And another. Search on drywall floating corner cracks for more references. And one in my comment on @bib's answer from the Gypsum Association; they might just know a thing or two about hanging drywall.