Yeah I guess there is not much you can do at home... someone needs to make a robot that can throw people! (preferably non-fatally)
Breakfalling when getting thrown is never going to be the same as breakfalling by yourself, because every throw is different, and you don't have a lot of control in how you get thrown :P. Also, if your opponent is taller and they hold on to your gi after they throw, it makes breakfalling much more difficult, because your back won't really be contacting the mat.
IMHO the perfect breakfall is really not that important. As long as you keep chin in, feet apart, and don't stick your hand out, you won't get injured. And in some cases, namely tournaments, people would actually deliberately not breakfall, because breakfalling and making a loud noise is basically telling the referee to give your opponent the point.
It would be cool to see cats do judo!
And yeah different clubs is good! That's the best thing about judo for me - you get free friends everywhere! And every club is a little different, with different focuses and different atmosphere (social vs hard core, etc).
Eg. I am from a small (~10 people) and casual club in Vancouver, and when I worked in California a while ago I joined a judo club there that happened to be one of the oldest club in the US with ~150 members and a huge gym - everything is so different! Made tons of friends, and had a send-off party to send a girl to 2012 London Olympics (she ended up winning bronze). I'm going to London in a few months, and will definitely find a club there to join, too!