Likely two separate issues. The backup error can be determined by looking at the LiteSpeed Activity tab in the console and seeing what error was reported (but you need to have a local repository to do this). If that doesn't reveal what the problem is, then you can turn on logging in the maintenance plan task by clicking on Reporting and Logging toolbar button in the designer and selecting the Create a Text Report option and examining that file after a failure.
The problem with the designer not properly showing the tasks does not mean they are deleted. There was a report of this same issue in an older version of LiteSpeed when a customer used Management Studio to try and edit the maintenance plan and then went back to LiteSpeed. I beleive the problem was solved in a more recent version - what version are you using? You could try exporting the MP and importing it back to see if that clears up the issue. If not, you may need to open a Support case for more assistance.
Thanks.