1 - Log out from Twitter from the app? I don't know what you mean.
2 - I haven't used ubertwitter in a while, and I know it's been upgraded since I used it, so I can't be sure. Check options from within the application and see if there is a setting to run in the background when closed, or something like that. Select help from the menu within the application and read.
3 - When in the options, you probably see a setting for update frequency. If you have set it to run in the background or you have it running, it will be operating according to the settings you have chosen.