Try to “Mount” the card first, there’s a button for that … here’s a guide (different device, but TWRP works the same).
And … last time I checked, TWRP doesn’t seem to support exFAT, so if the card is formatted in the exFAT filesystem, Android can use it just fine, but TWRP can’t.
Yes, you can do that (if there’s enough space left), afterwards you should copy the backup away from the phone anyway.
TWRP supports MTP, so you can just connect the phone via USB to a computer, and TWRP will show Internal Storage (and the SD card, if possible) for transferring files.