Are there specific halachos on books and assorted things being left on the bema after use ?

Shalom, Thank you for your question. Yes, there are halachot about what the objects that are designed for holy purposes can be used for. In general one is not allowed to lower the holiness of an object by using it for a profane usage, or even a holy use if it is of a lower level of holiness than the original use. Based on this we would imagine that the Bimah, or more accurately the cloth that rests on it, which is designated for the Torah to rest on, should not be use for lower level uses, such as putting books on them. However, the Rema writes in the Shulchan Aruch (Orach Haim, 154,8) that because there is a widespread custom, and it is very hard to be careful never to use these things for a lower level usage, there is a pre-existing condition on these objects that allows this use. By making a condition before the object is set aside for its use we have very wide sway in limiting the level of holy use attached to the object. When the custom is so widespread it is as of this condition was automatically made. So, in this case, the Bimah may be used for all normal shule uses - such as putting siddurim or sefarim on it, Parshat HaShavuah sheets, or the Tzadka box. One should though be careful not to put anything that is a disgrace on the Bimah, such as a newspaper, or a garbage bin (when cleaning the floor of the Shule for example). Blessings.