1) To access your internal HDD from an external source (work, parents) is a router issue and nothing to do with LE. There is nothing in LE (as far as I know) that can assist.
2) LE requires only what it needs on the SD card and memory, as the OS is loaded into RAM at boot. However /storage is allocated on the drive. So you can allocate /home to another drive if needed.
1) LibreELEC is designed as a htpc application, not as a cloud storage solution.
I'm just curious: a single 4TB drive with videos/pictures. Do you have a backup somewhere of all those files?
Because I sometimes ask people: what happens if such a single drive suddenly fails?
2) LibreELEC uses two partitions: One 512MB system readonly partition, and one storage partition which is automatically resized to the full size remaining disk size upon the first boot of LibreELEC.
You could set up whatever cloud-solution you like through Docker which is available as an addon.
If HDD die, I'm not sure what to do. Just like if DVD damage, it's gone. If u have better way to save family pic/video, please share.
Like I said... Make a backup. A simple copy of your files on a 2nd hard disk is a first step to keep your files safe.
There are two docker addons in the linuxserver repo you might be interested in.
Nextcloud will allow you to access your files remotely (it's like a self hosted Google Drive)
Duplicati will let you backup your files to the cloud or another computer running Duplicati
I'm lucky and found a USB port on my Linksys E3000. I configured Samba sharing, opened Kodi, typed in the local router's IP, and boom... I've got a "local cloud" 
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You can add a web address as a source in file manager.
You can add and FTP or sFTP as a source in file manager.
WebDAV server...
This will get you what your looking for if on the other end you have the software setup for FTP or Web access etc.
Many routers allow connecting a Hard drive to them. i.e. Linksys 1200AC has esata and USB3.0 and has the ability to setup sFTP for connection over the internet.
ALSO,
There is a OneDrive add-on under Video Add-ons that allows access to your Microsoft OneDrive Account. ---> How to set up and use OneDrive on Kodi | Windows Central or Add-on:OneDrive - Official Kodi Wiki
From what I can tell there is also a Google Drive add-on also, Help Google drive movies/TV files intergrated into Kodi
All solutions are Kodi based and not LibreELEC.
