Can I trust Apple support and share my password?
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.