Assuming a secured page https://google.com/redirect.php?keyword=apple which redirects the visitor to a non-secured website http://apple.com, would an eavesdropper (anyone watching over the network) be able to see the keyword "apple" from the referer URL?
In your specific example Google are hiding the
referer from click throughs from the results when Google is accessed via HTTPS, and have been for some time.
Non specifically, browsers should not send
referrer from https to http but it's possible that some browser implementations do - so the answer to your question is "probably not".
No. Browsers specifically block the
Referer header when the transition is from HTTPS to HTTP.
(HTTPS to HTTPS is fine, though following results from Google is a special case: depending on browser, interface type and wind direction, Google may be fiddling with the displayed URL and doing a search without a re-navigation, in which case you may only see
https://www.google.com/ and no search term even from an HTTPS site.)
Yes and no.
When you visit a non-secured site from HTTP over TLS (i.e. HTTPS), that non-secured site may get the referer line (depending on the web browser), which could contain a keyword from a search engine site.
However, in this specific example with Google, the actual keywords searched is hidden. Using Firefox, the referer line in your http request to apple.com might look something like this (tried this myself using Firefox 24.0):
Since the exact keywords used is not easy to determine, Apple.com's best source for information of this nature would be through the Google Analytics service.