There is a process that must be followed strictly if installing multiple versions. Install the oldest version first and then install all updates for it. Then install newer version and all updates.
Given that you have Office 2010 I cannot understand why you would want to install Office 2007 as it is by far the inferior version. There isn’t an issue with document compatibility so I just don’t see the need, unless you are a VBA developer and require it for testing client code.