Must I close the file first?
No, not unless you already have all 10 possible file entries used up, It depends more on your intention to write a 'clean' program or just want to hack your way ahead (*1).
How can I direct chrout to the screen?
The compatible way is by setting the device the following output is meant to go to. And setting it back afterwards. It's eventually exactly what the BASIC command
CMD does when redirecting output.
The function called is
$FFC9) with the logical file number to be supplied in X (3 in case of the screen).
*1 - Even though this was intended as a rhetorical question, there is always a chance to go brutal ... err ... use an inflexible low level approach. Here it would be using screen functions directly. Here outputting a char (in A) is