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.
Nit picking on terminology the roof (or the closet drywall) will be damaged. That's what cutting a hole is. What you need to concern yourself is will it be repaired properly.
Basically a roof consists of the roof framing, the sheathing and the weatherproof covering.
The framing should not be cut in the roof. It just complicates things too much and isn't necessary. If it accidentally does get cut it can be repaired.
The sheathing depends on when your home was built. Decades ago you may have seen solid wood boards such as 1x6's, later plywood and more recently either plywood or OSB. Plywood and OSB are easier to replace and more people will have experience with it. If it's an older home with solid boards as the decking you may want to avoid messing with it.
The covering, things like asphalt shings are fairly easy to replace. If you have something like a clay tile roof then this would be more difficult but I doubt the contractor would have suggested cutting through the roof if that were the case. I think it would be best to use new shingles when repairing the hole but I'm not a roofer. With new shingles you'll have a patch that looks different than the rest of the roof.
Cutting through walls is similar in that there is framing, drywall then drywall compound over seams and finally paint. Much easier to repair and if something goes wrong with the repair you don't have to worry about a leak.
Going through the interior closet is the safer choice for you, going through the roof probably makes attic access easier for the contractor and could be less messy for you since all the cutting and running of insulation hoses happens outside.
Since your walls are plaster it could make the repair more complicated but the repair can also be made with drywall, shims and maybe a little extra mud to even things out. It may not come out as could than properly repairing the plaster but in a closet I wouldn't mind too much myself. The other option is to cut out a proper sized attic access and install a hatch. To be code compliant the opening is larger than the 16" OC the ceiling framing probably is (I believe minimum is 24"x30" but double check). That would involve adding some framing around the hole to accommodate it. Not that big a deal but might be something to consider if you'd like to have access to that attic area in the future. The larger hole should also make it easier for the contractor to bring up insulation.
I personally would feel more comfortable with cutting a hole in the drywall than the roof. If you like the contractor/equipment/insulation from the guy that wants to go through the roof you can probably tell him you'd rather he go in through an interior wall and not make any holes in your roof. He'd probably agree to it.
Best Answer
First you need to lay out the general area where you want the opening. It needs to fit between 24" centered ceiling joists.
Properly installed electric cabling is always at least 1 1/2" away from the back edge of the sheetrock (or protected by a metal shield). So, if you cut a small hole near the middle of your proposed opening with a shallow cutting tool, such as a utility knife or a multisaw, big enough to get your hand and arm through, you can reach in to make sure there are no wires draped in the area where you are working. If clear, cut a bigger hole with a drywall saw or multitool, large enough to get your head through. Insert a light and then your head and scan the area.
Now you're prepared to cut the full opening back to the joist with no surprises.