Uploads and downloads can continue without hosts showing on the Connections page (the tranfers are taking place just like transfers from the web via http), so that might be why you see downloads resuming without the connection light being green.

You need connections to the gnutella.net in order to relay search queries and to initiate downloads, but if your light is green, that means you are connected and should be able to search. Check View->Show/Hide->Connections to see the connection page. It should show you connected to 4 (6 if you use Pro) hosts. If the hosts are showing then disconnecting quickly, there might be a problem with your modem or further upstream.

If you have stable connections, the searches should work. Is it just that the search button is greyed out?