Cadabra's philosophy is that notebooks should be viewed as code, which you should be able to run from start to end in one shot if needed. This clashes rather strongly with the idea of cutting and pasting (parts of) expressions which are then manipulated in following cells: the information about what you cut and how that should change when any of the earlier cells change is not contained in the notebook. This is very, very bad for reproducibility; try to avoid working like this.
To enforce a better working style, there are various tools in Cadabra to select sub-expressions programmatically and then act with algorithms on only those sub-expressions; I refer to the section on zoom/unzoom in the manual: selecting expressions
Re-arranging expressions by moving terms to the left/right of an equals sign should be done using to_lhs and to_rhs of the cdb.core.manip module. In this way, if you expression changes, you are still sure to do the right thing, and you can run your notebook without doing the cut-n-paste again.
Now having said that (which addresses part 1), I am sympathetic to part 2 of your question: sometimes you simply want to copy a result into a LaTeX file somewhere else. There is no cut-n-paste way to do that, but you can use the pyperclip python module to put the LaTeX form of an expression into the clipboard (and then paste that elsewhere):
import pyperclip as pc
ex:= A_{m} + B_{n} D_{n m};
pc.copy(ex._latex_())
You can then paste the LaTeX form of ex to another program.
Of course you can abuse the above to cut an expression, edit it elsewhere and paste it back into a new input cell, but I would advise you not to do that, and learn instead about zoom/unzoom/to_lhs/to_rhs and various other tools.
Hope this helps.