[T-Mobile Samsung Galaxy Note 3] Apo11on:
I believe the problem can be seen starting line 163 in the above recovery.log file: /external_sd | /dev/block/mmcblk1p1 | Size: 0MB Used: 0MB Free: 0MB Backup Size: 0MB

Shows the same zero size I see in TWRP UI.

Here is the full output of the mount command from inside my KitKat rom, where this microSDXC is properly mounted (I think I highlighted the line that's relevant):

=/data/local/bin:$PATH <
u0_a230@hltetmo:/ $ mount
rootfs / rootfs ro,relatime 0 0
tmpfs /dev tmpfs rw,seclabel,nosuid,relatime,mode=755 0 0
devpts /dev/pts devpts rw,seclabel,relatime,mode=600 0 0
none /dev/cpuctl cgroup rw,relatime,cpu 0 0
proc /proc proc rw,relatime 0 0
sysfs /sys sysfs rw,seclabel,relatime 0 0
selinuxfs /sys/fs/selinux selinuxfs rw,relatime 0 0
debugfs /sys/kernel/debug debugfs rw,relatime 0 0
none /sys/fs/cgroup tmpfs rw,seclabel,relatime,mode=750,gid=1000 0 0
none /acct cgroup rw,relatime,cpuacct 0 0
tmpfs /mnt/secure tmpfs rw,seclabel,relatime,mode=700 0 0
tmpfs /mnt/secure/asec tmpfs rw,seclabel,relatime,mode=700 0 0
/dev/block/vold/179:69 /mnt/secure/asec vfat rw,dirsync,nosuid,nodev,noexec,noatime,nodiratime,uid=1023,gid=1023,fmask=0007,dmask=0007,allow_utime=0020,codepage=cp437,iocharset=iso8859-1,shortname=mixed,utf8,errors=remount-ro 0 0
tmpfs /mnt/asec tmpfs rw,seclabel,relatime,mode=755,gid=1000 0 0
/dev/block/dm-0 /mnt/asec/com.probosoft.markethistorydelete-4 ext4 ro,dirsync,seclabel,nosuid,nodev,noatime,errors=continue 0 0
tmpfs /mnt/obb tmpfs rw,seclabel,relatime,mode=755,gid=1000 0 0
/dev/block/platform/msm_sdcc.1/by-name/system /system ext4 ro,seclabel,nosuid,nodev,relatime,errors=panic,data=ordered 0 0
/dev/block/platform/msm_sdcc.1/by-name/userdata /data ext4 rw,seclabel,nosuid,nodev,noatime,discard,journal_checksum,journal_async_commit,noauto_da_alloc,data=ordered 0 0
/dev/block/platform/msm_sdcc.1/by-name/cache /cache ext4 rw,seclabel,nosuid,nodev,noatime,discard,journal_checksum,journal_async_commit,noauto_da_alloc,data=ordered 0 0
/dev/block/platform/msm_sdcc.1/by-name/persist /persist ext4 rw,seclabel,nosuid,nodev,noatime,discard,journal_checksum,journal_async_commit,noauto_da_alloc,data=ordered 0 0
/dev/block/platform/msm_sdcc.1/by-name/efs /efs ext4 rw,seclabel,nosuid,nodev,noatime,discard,journal_checksum,journal_async_commit,noauto_da_alloc,data=ordered 0 0
/dev/block/platform/msm_sdcc.1/by-name/persdata /persdata/absolute ext4 rw,seclabel,nosuid,nodev,relatime,data=ordered 0 0
/dev/block/platform/msm_sdcc.1/by-name/apnhlos /firmware vfat ro,context=u:object_r:firmware:s0,relatime,uid=1000,gid=1000,fmask=0337,dmask=0227,codepage=cp437,iocharset=iso8859-1,shortname=lower,errors=remount-ro 0 0
/dev/block/platform/msm_sdcc.1/by-name/modem /firmware-modem vfat ro,context=u:object_r:firmware:s0,relatime,uid=1000,gid=1000,fmask=0337,dmask=0227,codepage=cp437,iocharset=iso8859-1,shortname=lower,errors=remount-ro 0 0
/data/media /mnt/shell/emulated sdcardfs rw,nosuid,nodev,relatime,uid=1023,gid=1023 0 0
tmpfs /storage/emulated tmpfs rw,seclabel,nosuid,nodev,relatime,mode=751,gid=1028 0 0
/dev/block/vold/179:69 /mnt/media_rw/extSdCard vfat rw,dirsync,nosuid,nodev,noexec,noatime,nodiratime,uid=1023,gid=1023,fmask=0007,dmask=0007,allow_utime=0020,codepage=cp437,iocharset=iso8859-1,shortname=mixed,utf8,errors=remount-ro 0 0
[COLOR="Red"]/mnt/media_rw/extSdCard /storage/extSdCard sdcardfs rw,nosuid,nodev,relatime,uid=1023,gid=1023 0 0[/COLOR]
/data/media /storage/emulated/0 sdcardfs rw,nosuid,nodev,relatime,uid=1023,gid=1023 0 0
/data/media /storage/emulated/legacy sdcardfs rw,nosuid,nodev,relatime,uid=1023,gid=1023 0 0
u0_a230@hltetmo:/ $