As to a file system being mounted read-only, you can try to "re-mount" it read-write in this fashion:
mount -o remount,rw /your-directory-name-here
As to the potential LVM related issues, you should make sure that you have logical volumes,
formatted to the same size as you've told the LVM they are ( in the sense of resize2fs ) , mounted
in the right place and such. For example on my system, in what you've shown us, if the logical
volume you're showing us in what you've posted, was meant to be a part of the volume you've
shown us, the VG name field for the logical volume would be the same as the VG name field
in the volume group information. Are we missing some output here?