I'll speak to the pull-down stairway and leave advice about other insulation to those who know. There are manufactured products designed to cover stairway from above but I installed a budget job. I purchased 2 sheets of foam board insulation from Lowe's. I cut 3 rectangular shapes slightly larger than attic stairway hole framing, bonded them together with construction adhesive in a stack and laid them atop access dor framing members. Yes, now when I pull down stairway I must push up the foam board insulation and lay it aside in attic before walking up. It is a inconvenience I can live with. With smaller pieces of foam board left over, I measured and cut strips that fit backside of pull-down stairway between plywood face and stairway rails plus between treads. Finally, I applied a thin weatherseal strip on lip of access door. Nothing unsightly. I invested about $20 and 1 hour.

The biggest question is...

Just how much of the loose gray stuff (cellulose insulation) is up there right now?

You need to know the depth in inches. When you measure, just make sure you're not on top of one of the joists, you need to be down against the ceiling drywall.

In North Carolina you should have enough insulation to bring you to an equivalent value of R-38, at a minimum.

Cellulose, even settled, has an R value of about 3.6 to 3.8 per inch.

For argument's sake, let's say 3.5. In order to meet the R-38 value, you'd need about 11 inches of insulation.

Weather stripping the hatch is always a good idea. The less air infiltration you have into unconditioned parts of the house the better. Hikker's solution is a good one, but there are also insulation "blankets" that you an install over the hatch opening that do the same thing.

Generally it's easy to tell if you have insulation in your walls.

At your circuit breaker box, turn off the power to an outlet on an outside wall and move the cover plate and switch. You should be able to see if you have insulation or not.

Increasing the insulation in the crawl space probably isn't necessary.

1. What kind of insulation and how much should I have put into the floor of the attic to bring it up to standard?

Add insulation to make it R-38. blown in over the cellulose. If you can see the ceiling joists you need insulation!

2. Aside from making sure that vents in the floor of the attic that provide air flow are not blocked, what else should I look out for?

Areas that leak air and transfer attic temps into the living/conditoned space. This can be oversized cuts around bath vans, kitchen vents, fireplaces and recessed lights.
(to name a few) If you can make as complete as possible
air barrier, then you can add additional insulation.
The question then is..where to seal. Blower door test.
Have the person you are going to hire to seal these areas
there for the test.

3. We have a vaulted ceiling in the kitchen and living room. We were told by one contractor that blown in extra insulation in those areas is a good idea. Another contractor said not to bother.

I think that your vaulted ceiling is what I call a cathedral ceiling. (one is more of a triangle than the other..) adding insulation on these slanted surfaces can be done, but again, after sealing all penetrations to the living space. The end walls (knee walls) of this area should also be sealed. I ususally use a 1' foam/foil sheathing board on top of the insulation in the knee walls, batt insulation is the only kind of insulation that can somewhat easily be installed, foil/foam board is installed with button cap nails. the foil reflects the heat back into the attic while the sheating board stops air infiltration and heat gain.

4. We have insulation under the floor -- batting material in the ceiling of the crawl space. It's about 3 inches thick. Is there any value in increasing the thickness of this insulation? The hard wood floor are not cold to the touch in winter.

Other than pulling it out and putting a couple of inches of foam insulation..I'd prolly leave it alone.
You might seal leaks in the floors, pipe penetrations and such. Usually floors leak at the floor to wall junctions.
In the years I have been in my house I have caulked all the base boards to the walls, and to the floors..same for the ceiling moldings. Stopping air infiltration makes it easier to heat and cool your home. Insulation slows down air flow, but it doesn't stop it. When air moves through insulation is robbs it of its insulating value. When air moves through the insulation into your home, it brings moisture, insulation particles, and ambient and higher temps.

5. The siding is made of that Masonite product that got a bad rap a few years ago. We have looked it over very closely and, after replacing it in a few places with Hardiplank where water damage got to it, we feel confident that the rest of it is okay. We understand that it's insulating capacity is at least as good as wood if not better. We think there's insulation in the walls, but don't know how much or what kind. We don't plan to do anything with that any time soon. Is that sensible?

Again blower door test will tell you of the areas that leak. Measures and identifies air leakage areas.
If you don't have insulation in the walls, holes can be drilled and insulation blown in, a hole at the top of each wall cavity and possibly one halfway down ..if the walls have fireblocking. Holes are then capped off with plastic caps.

6. We are replacing most of the window sashes with Energy Star rated windows, and are putting in new weather stripping where needed.

Good deal! Low e..vinyl? Solar Heat Gain Coefficient
SHGC .35 or less?

7. We have pull-down stairs to the attic. Is there a reasonably priced, sensible way to insulate the hatch? Or should I just put weather stripping there?

Products like attic tents seal well.
building a box as described above works well also, as long as it is airtight and seals well. I find that they have to have a little weight to them to acheive a good seal.

Best of luck to you!