Android Hold down the desired conversation from the main conversations page. Select 'yes' when asked to confirm. Iphone Swipe right and tap delete on the main conversations page.
To search inside a chat simply tap the search icon and enter the text you wish to search for. There are arrows to toggle between different instances of the located term and an "X" to close the search.
No, you will need a contact with either an email or number in order to initiate conversations. An alternative is to have that person visit your Digital Business Card and start a chat with you.