Vinyl siding over old siding

I REALLY don't like doing things twice ...

This might indicate it is not a DIY project for you. However, if you are willing to do a few bits and pieces twice, you might be able to master the work well enough for a permanent result.

I also hate paying 3-4x more for someone else to do it.

This is often the best motivator, and hopefully provides some perspective. If you end up spending 20% more than a perfectly executed DIY, you are still way ahead.

1, should I do it (hardy) or the cedar

Dunno without seeing it. My house is brick on one face (the north side, oddly) and cedar shingles on the other sides. As long as the finished product looks like it is supposed to be that way, it should be fine. Have you thought about planting shade trees on the south side?

As far as redoing the siding's cladding/sheathing, you probably won't know until after tearing off the siding. It could be that part was well done. I don't think it needs exterior rating because it should be well protected and never in contact with moisture. 1/2 inch plywood or chipboard is standard, but thicker won't hurt and will help a bit with sound insulation and overall durability.

As far as house wrap, brace yourself: it is surprisingly expensive (maybe $100 for one side). If airflow through the wall is a significant factor, like all the other walls are tight and this wall is the big leaker, then it is worth it. If all the other walls and windows leak significantly, then don't bother unless you plan to eventually seal them all.

You might also consider adding additional insulation while that is open, both inside the wall cavity and perhaps as another layer on the outside. This might cost thousands, but given the high price of electricity there and typical air conditioning use, it probably will pay for itself in 8–15 years, as well as being a big improvement in comfort.

If you are going to replace windows, it will be way easier with the siding off than at any other time. The flanges are screwed into the cladding from outside. 14 years isn't all that old of a window, and a straight replacement isn't likely to save much. However, an upgraded replacement could well save on energy costs. Again the pay-for-itself period is probably in the 8–15 year range. Get an energy audit from your local utility (usually free) to learn more.

When repairing a small section of siding, it's a best practice to tape the holes and rips in the weather wrap (which you refer to as "paper") with the appropriate tape. But for a larger area, it's faster to simply replace the entire section of weather wrap and I would insist on it. Nail holes and tears compromise the effectiveness of the material, so I wouldn't recommend leaving it on when residing the house.