Steven Zahm@ Helen
RE: Can I use the search facility on the basic template (which I usually slightly restyle and reorder)?
If you use the Search widget included in the Widget Pack the Default template will work correctly with it. That said, I would recommend the cMap template instead (the Searhc widget also works with cMap if you want to use both).
RE: Can I customise it to find information in the new fields I have made?
If they are text field, yes. All you need to do after you add the fields is navigate to the Connections : Settings admin page and click the Search tab. You will find your newly added text field as search options. Enable them to have search query those fields. I would recommend keeping the number of field to a minimum as to keep the results as relevant as possible for your users.
RE: Can you place a search box for Connections in a widget (sidebar or footer)?
Yes, the Search widget can be placed in any widget area that the theme supports.
I hope I have thoroughly answered your questions!