Unable to Share photos through Email in Photoshop Elements

I have no difficulty setting my hotmail (or outlook) profile in the organizer.

The process does not validate "hotmail email addresses", it validates your own hotmail account to be able to use your hotmail service with your id and your password. I just did it again. You choose 'Microsoft' as the type of account and enter your id and password as if you you opened your hotmail account. No complicated two steps  validation as required by Gmail. Once your hotmail account is recognized, you enter the addresses you want in your contact book.

I don't have a Hotmail account so can't give any advice about the specifics.  But some email providers require extra layers of security to enable the use of third pary applications to manage emails.  For example, gmail makes you jump through numerous hoops to get a separate password to enter into Elements email.  See this thread.  Perhaps it will give you a clue to some settings in your Hotmail account.

What OS and version of Elements are you using?

Until you are able to resolve the issue, you should simply use your email client to attach or insert the photo(s).