"..how do they keep the leaves out of the cup?"

I have no idea how most Irish do it, but there are many devices to use loose tea... even very fine tea. I have several 'tea balls',,,including a couple of 2-part plastic things with a fine mesh on the inside. Tea steeps until YOU are satisfied, and no leaves.

I have no idea why people put milk in tea,,, except possibly because they got stated using cheap tea, and that was the only way to make it palatable. Now it has become a cultural thing, even when top-notch loose tea is available.
(I knew a couple in college many years ago who were VERY frugal getting thru college. They made VERY cheap tea VERY strong, re-used the tea bags, and put lots of milk in it. I suppose that made it healthier, also)