What can I do to alleviate this and are Sony cameras particularly prone to this problem?
To the best of my knowledge Sony are not renowned as being more prone to the problem. The problem itself is so common, and I suspect the red-eye reduction routines on most camera so ineffective, that most if no all common photo-editing programmes carry some form or red-eye correction routine - in some cases these are so highly automated and effective you need do little other than identify the guilty pictures and let the software do the rest.
Quote: Are there any makes which are not?
Not that I know of. Best option is to look for models where the flash is as far from the lens as possible to minimise the effect. Even my digital SLRs will produce red eye and the pop up flash on those is further away from the lens axis than any compact.