Connecting toilet to water supply

[Plumom]

I brought a HUD foreclosure and I need to repipe the entire house because the copper piping was stolen. I'm going to plumb the entire house using PEX. There's a remaining piece of copper pipe left behind where the toilet's water supply pipe would of connect to that piece I can't think of the name which attaches to the underside of the water tank. I don't know what I would need if I had to resemble this entire section. Can I heat the remaining copper pipe to remove it and install a new 1/2 inch pipe in its place to make the connection to the toilet? The same thing goes for underneath the bathroom and kitchen sinks. I'm pretty certain I have a good enough length though of copper pipe underneath the kitchen sink to connect it to 1/2 x 3/4 inch straight coupling (Sharkbite) to connect them to the 3/4 inch Pex pipe I'm going to be using underneath the house.

[phishfood]

I am having trouble visualizing the situation. Could you attach some pictures with notes, or a more indepth explanation of what you are working on?

[havasu]

I'm also confused as to what you are asking but make sure each toilet and sink has dedicated shut off valves. From that point, you should use a stainless flexible supply line from the valve to the faucets or toilet.