That's odd. Try Solid Explorer. That has been my main file explorer since it came out, and it worked for me on stock.
Sorry, but I'm pretty sure this is wrong. SD Card access is fixed in Lollipop, but, again, it's up to the app developers to update their apps to the new API. Also, I'm pretty sure all phones have kernel R/W access to the SD Card, or at least FAT 32 & SD Card drivers.Even in KitKat, this was limited by an Android permission, not the kernel.
Just want to warn u about ExFAT. ExFAT support in Android & Linux is spotty, at best. Only a little bit of android phones support ExFAT, I'm not even sure if this one does. Even if stock happens to support ExFAT, CM definitely won't, and if u ever decide to move to another phone you'll have to format it.
System Image is a new feature in TWRP that makes a 1:1 fastboot backup of the system partition. It was made because Lollipop updates require the system filesystem to be exactly the same as when the phone was shipped. This feature was made for phones that use fastboot, and thus don't need to root before flashing TWRP. It's completely useless to us.