What size of ceiling joists should I use
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.