I'm not really getting what you're asking, usually type L is required for domestic water (I'm not sure it's code or not but in these parts they don't allow thinner "M" for domestic water use). We can only use thinner type "M" in heating systems.
As for what size (1/2" vs. 3/4") the issue is the speed at which water flows through the pipe. Once you start reaching over 8FPS the water moves so fast through the copper pipe it starts to tear away the pipe inside causing the pipe to fail much sooner.
That being said, the normal rule of thumb is 1/2" pipe can supply up to 3 fixtures in the worst case scenario (all 3 fixtures at 2.5GPM turned on full at the same time) and still keep things under 8FPS. If your bathroom has a shower/tub, toilet, and 2 sinks you need 3/4" pipe on the cold side (shower, toilet, 2 sinks = 4 fixtures) and 1/2" pipe on the hot (shower + 2 sinks = 3 fixtures). If your bathroom has only one sink 1/2" should do for both hot & cold. Everything in a bathroom wants 1/2" pipe, including the shower head, bathtub, toilets, and sinks. If you do have 4 fixtures usually it's just 3/4" pipe into the bathroom and from there you add a 3/4" x 1/2" x 1/2" tee and run 1/2" pipe to each fixture. I don't believe it's code so, even if your bathroom did have 4 fixtures I don't believe anything's stopping you from using 1/2" to supply it... it's not likely someone will be running both sinks on full cold while the toilet is flushing and someone's taking a pure cold shower/bath. It does happen though, getting rusty water it's common for the person to turn on all the fixtures of a 4 fixture bathroom at full cold while flushing the toilet to clear it... that situation will massacre the 1/2" copper pipe supplying the bathroom and better you used 3/4". 3/4" pipe has twice the volume of 1/2" so it can supply up to 6 2.5GPM fixtures turned on full blast at the same time simultaneously before you have to worry about 8FPS being reached.