Yes, all are correct. You can even add "to have a coffee" to the list. I think that's used more frequently in the UK but it is common in the US as well.

Coffee is not something we count. Meaning we don’t say “I had 3 coffees today.” For things that we count, like meals for example, we say “I had 3 meals today.” So in the situation with coffee/tea, we would use the first and the third examples you gave: “To have a cup of coffee/tea.” “To have some coffee/tea.” That is a general rule you can keep in mind for other liquids as well, like milk and water. Hope it helps.