@jdbavon
On your "1-on-1" list, under the little chat bubble is a small "x" icon. Clicking that removes it from your 1-1 list, unless they message you back about something, in which case you'll get a notification. On the top right of the chat window is a small bin icon, clicking that will delete all the messages from the chat, so it is then emptied, if that's what you meant too.
As for the App, it's a work in progress. If you're noticing technical issues with it, please report them so they be fixed and improved. It is currently behind the site btu it's being heavily worked on, sorry to hear you're experiencing such issues with it currently.
Hopefully that answered your questions, take care.
~ Baubs