[Sian O'Brien BA MCP]"VLC Codec Information states they are 720x608, whereas MediaInfo says they are 720x576."
I think VLC does not support "display area" info from e.g. MXF, so displays the whole frame, including VITC in VBI. Not a problem for broadcasters if they support such kind of file (you need to ask them, but they usually do, this is classic)
If you look at text of tree view in MediaInfo, you should see both values (608 and 576).
[Sian O'Brien BA MCP]"They're also supposed to be 16x9, which I thought was 1024x576"
Only in the case of square pixels. You can not suppose of width with only height and display aspect ratio, corresponding width depends of Pixel Aspect Ratio. For PAL (720x576) this is **not** square pixel aspect ratio.
[Sian O'Brien BA MCP]" If these files are 720x576, does that mean they're anamorphic videos? And if yes, will this be a problem?"
Yes it is anamorphic, no it will not be a problem because it is standard PAL (you would have a problem if the file is not anamorphic, actually)