I use FreeBSD, is it affected?
Yes. I probably should have said "unix-like". The issue affects any system where you're dynamically linking Bitcoin Core against the system OpenSSL. This could even be done on Windows, though that'd be very unusual.
Yes.
Is the issue having the new version of OpenSSL at compile-time, or at run-time? (My build of 0.10rc1 links dynamically to /lib/x86_64-linux-gnu/libssl.so.1.0.0, but I don't know about 0.9.3 or builds made on the PPA as part of a Debian build process).
It's an issue of what Bitcoin Core will use. If it's statically linking an OK version of OpenSSL, then updated your system OpenSSL is OK. If it's dynamically linking, then you'll have problems. The binaries on bitcoin.org statically link OpenSSL. I think that almost all Linux distros distribute versions of bitcoind/bitcoin-qt that dynamically link.
If you're compiling Bitcoin Core using the normal configure+make, then it'll link dynamically. I'm not sure how to force this to link statically.