Assume you have a piece of paper.
Now you if put a pencil on the paper, you can say: 把一支笔放在纸上。As you've already known, 放 just means "put", so there is a wide range of usage.
If you put a stone on the paper to avoid your paper being blown away. You may want to emphasize the weight that the stone put on the paper, you can say: 把一块石头压在纸上。
If you put another piece of paper on the paper, you can say: 把一张纸叠在另一张纸上。You usually 叠 something similar together. say, 把桌子上的纸叠到一起，把书叠整齐，etc... Usually they are thin stuff, but there is an exception, that we call human pyramid as 叠罗汉 (罗汉 means people here)。
If you put something on the paper (e.g. a hat), and want no body see what you write on the paper. You can say, 把帽子盖在纸上。
If you put the paper on water, the paper floats on the water. Then you can say, 纸漂在水面上。If you want to emphasize that, its density is low, so that if you put it down to water, it can come back to the surface. Then you may say, a rubber duck 浮在水面上。
Other cases may depends on what the action look like, then you choose a corresponding verb: 衣服挂在衣架上，气球悬在屋顶上，刀子架在脖子上 etc.
You can use all of them with "着"：纸上放着笔，压着石头，叠着书，盖着帽子，supprisingly with all these stuff, 它还在水面上浮着。