Apple sends a text message, others suddenly don't display the note, and the recipient becomes an iCloud address.

Just ask her to add your icloud account to the address book.

Because after you change your appleid password, imessage requires you to log in again. When logging in again, you will choose the default mobile phone number or email address. Maybe you chose an email address.

Of course, if you want to completely change it, you can click Settings-Messages-Send and Receive-and use it when starting a new conversation, just choose your mobile phone number.