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.

Safety first

Make sure you wear proper safety gear. Safety goggles, appropriate gloves, hearing protection (if applicable), and a hardhat, might all be useful safety equipment.

Watch out for that...

Make sure you know what is inside the wall/floor, cutting through a wire or pipe can ruin your day really fast.

"Did it come out yet?"

It might be a good idea to line the chute using ducting or other lining material, so the items you drop in actually make it out the bottom.

Don't bring down the house

The easiest and safest place to install a laundry chute is between studs/joists. If you have to cut studs/joists to install the chute, you'll have to reinforce the structure appropriately. If you are going down more that a single floor, you'll likely have some structural issues to deal with. If you go between studs, you'll have to deal with a sill plate at the bottom of the wall and possibly a top plate at the top of the wall below.

The right tool for the job

Make sure you have all the proper tools before you begin. Having all the right tools you need at hand, makes every job easier.