Are we allowed to omit 载 when it's clear what we download?
1) Are 下 and 载 separable? I mean, are they kind of "verb+object" word?


No, '下载' is a compound word verb, you need both characters for the specific meaning of 'to download'


  (向)下 = downward 
  
  载 = to load onto
  
  下载 = to download (from upstream server to local storage device )


~


  (向)上 = upward
  
  上载 = to upload (from local storage device to upstream server)- It is also a compound word verb


~


  2) Can we omit "载"  if the thing we download is clear enough to the listener? For example, can I simply say : 下照片/下视频?


Yes, just like you can omit 'drink' in 'drink down' and write 'down a bottle of wine', presume the listener could reliably guess what the omitted word is. 

We don't "吃下" (eat down) a bottle of wine, only "喝下" (drink down) a bottle of wine.

However, you should always use the whole word 下载 or 上载 to avoid confusion. Because without other context, 下照片 could be interpreted as '把照片下架' (take down the photo) and 上视频 could be interpreted as '把视频上架' (post the video)


  3) If no. 2 isn't the case, then is there any specific rules/conditions of when we are allowed to omit "载" and when we are not?


You can omit "载" or any character in a compound word if you think the listener could reliably guess what the omitted word is; You cannot omit it if you think the listener could not reliably guess what the omitted word is


  omit 下 and write 载照片 or 载视频 only create more confusion. ('载 upward' or '载 downward' ?)