I 've bitten the bullet and now uploaded PS3 to my live (remote) server and will use the local server to update and change files as I need to.
The interesting thing, from my point of view, is that viewing PS3 on a live server seems totally different from viewing it on a local server. Or it certainly is in my case.
For example, since putting the categories into the store through admin on the live server, I no longer have this problem of categories duplicating themselves and all search links function OK.
Also, the problem I experienced with the menu - L1 overlapping L2 - does not happen when the site is live. Just as you predicted.
Why there should be these marked differences I couldn't even begin to guess. Anyway, things now seem to be sorting themselves out, so fingers crossed that I won't need to disturb you and your colleagues too much in the future. Thanks.