I am developing my basement that will need plumbing for 1 1/2 baths,
a basement kitchen, the washing machine, the laundry tubs and a wet
bar. My house has half inch copper pipes for plumbing. I like the
new fangled plastic piping that can run the whole (30 ft?) length
without a soldered joint.

PEX comes in rolls upto 250' so that will work for you.

All the plastic piping will run from a copper header, that thingy
shaped like a sausage and has 8 (or 12) 1/2 inch nipples for the
plastic pipes to connect to.

The "thingy" is called a manifold.
I'd like to connect the (Hot-Cold water supply) headers to the hot and the
cold water faucets that are
currently used for the clothes washer.

What size are your washer supply lines ? The normal supply for a washer is
usually 1/2". Your basement project looks like it will need 10 or more runs
so a 1/2" supply probably wouldn't be adequate. My place has a 3/4" supply
line for 11 runs. That said, it seems plain you aren't planning on getting a
permit or having an inpector/plumber check/do any of your work. Consider
what might happen with a basement leak flooding your basement if you DIY.

As far as connecting your new plumbing to existing washer faucets, you'll
get plenty of advice telling you it isn't code or even a practical way to
accomplish what you're trying to do. You should go right back to your main
supply where it comes into the basement and run a line for the new work off
a larger (than 1/2") line after the regulator.

Stand by for the pro plumbers replies ..