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.

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. 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. That way I don't have to cut
into the copper pipe to solder on a copper Tee to hook up a header.
The current washing machine faucets will also serve as shut off valves
for the basement pipe installation.

This is the question. I will be affixing the headers under a laundry
room cabinet to organize the plumbing and look tidy. The header will
be 5 to 8 feet from the supply faucet. I asked the Home Depot guy for
a female hose fitting that will let me connect the header to the
washing machine's faucet via plastic pipes. He couldn't help me. I
looked through their fittings and didn't find anything suitable (1/2
in plastic pipe with a female hose fitting.)

One alternative may be to remove the faucets to expose the 1/2 inch
female brass pipe elbow. I think I may have seen a (30 inch long)
plastic pipe section that had a 1/2 inch male pipe end to fit into
this. But I would like to keep the faucet as a shut off valve. 30
inch is too short anyway and if I have to use pipe connectors the cost
and complexity increases.

Any suggestions.

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 ..