Can I trust Apple support and share my password? I have problem with iCloud backup on my iPhone. After several calls to the Russian department of Apple support they suggest me to change my AppleID password temporarily to an test password proposed by the support specialist to see what's going on with my account. Also they say that they get access to all my data stored on phone: messages, photo, apps data etc, and I should agree with this terms. 

Of course they say usually engineers don't read/watch users data, but  I think it's weird to grant access to all my data. 

Should I trust them and share my pretty photos, banking apps and personal chats? I'm pretty sure I have talked with the official Apple support, not scammers.
In a word, NO.  No-one reputable will EVER ask for your password, EVER.  Proper support have the tools and such in place to allow them the access they need to do what they need.  If they have to actually login as you (Which clearly they can only do from a different device anyway, limiting any usefulness it may even have) then surely there is nothing they need to do that for which they can't simply ask you to replicate for them without handing over credentials.  I smell a rat.