but this text is green
Because you must change layer you put text into, for example, to "Info".
and there is no way to change it's size/font/etc
After you place text into schematic, you use info tool on it, and you should be able to change size, ratio and font type in that window. Actually absolutely the same as in board editor.
So if texts are not transferred between schematics and pcb, why is it matter to set the layer?
Because each layer is having its purpose. If you put text into Nets layer (not sure it is correct doing so), EAGLE will treat it as signal, which may need to get wired at the board level (again not sure, never tried this extreme case). Next, you can turn specific layers on and off, it is good practice to put appropriate objects in respective layers so that you can property visualize them.
I did and it really changed the color from green
There's Layer settings icon, you can turn layer visibility in that window, and you can see default layer color there. In general your text changed to gray because you moved text into another layer which identification color is gray.