Al,
Wow. Yes, I'm constantly amazed by some of the "corner cutting" practices used by some in the industry. Some of it just makes you shake your head.
Installing new windows at the same time will be the best way to go. As each window is removed, you will install your Tyvek (or similar) over the opening. The new window will get nailed on top of the building paper, with one exception- and that is the top. Before the window is installed, you cut a flap in the Tyvek on top and fold it up out of the way. That way when you install and tape the edges of the window, the tape will cover the nailing fin and tyvek on the sides, but on top it will be applied over the nailing fin and then directly onto the sheathing. After that is done, the flap of Tyvek is folded down over the flashing tape and then you can either skip tape it or continuous tape it, (depending on whether you are going for an air-tight installation or not.)
When you side, the j-channel and siding can be installed right up to the window. But some people feel that looks a little anemic, without any trim. So you can always add trim around the window. But with vinyl siding, you probably want something that is low maintenance. Azek #908 brickmould would be one option. It is supposed to hold paint well. Azek also comes in 3/4, 5/4 (which is actually 1") and full 1 1/4" thick (1 1/4" is known as ATM) and can be cut to any width just like wood. But it's pricy. You could also use standard wood #908 brickmould and have it capped with aluminum trim coil which also looks good. Unless you get lots of hail.

Sills can be fabricated and can be attached on bottom to make the windows with nailing fins look a little more like the style of window we are used to seeing. Any exterior trim you choose will be purely for looks. But you likely want it to be low-maintenance.
There are also some mouldings that can be applied around windows after the vinyl siding is installed. They basically hang on the vinyl siding like a shutter would, and just give the windows the "appearance" of having trim around them. Some of these mouldings come in fancy profiles, such as if you wanted the home to have a colonial look, with an entablature, cornice and such. So that's something else you could consider if you wish.