I have only competed twice but there were a few major changes that I made between the two that drastically improved my cardio (and actually helped me win gold in my nogi category):
1. It may sound odd but whilst waiting in-between fights I have found that listening to ambient/chilled music as opposed to aggressive music helped me remain focused without becoming too emotionally attached to the situation. I think this helps because when I fought in the second tournament I found it far easier to manage my adrenaline, subsequently preventing me from gassing too quickly.
2. EAT! As soon as you have weighed in make sure you eat plenty of carbs to keep you going throughotut the day. For example, before my first fight in my second tournament I ate a banana, a snickers bar and a sandwich around 45 minutes before I fought and it provided me with the energy I needed.
3. With regards to doing some form of cardio outside of jiu jitsu I would say definitely, however, nothing prepares you for jiu jitsu cardio like actual rollling. So, if you can do cardio outside of jiu jitsu make sure it replicates a fight. For example, I have a tournament coming up this Saturday and I have been doing 5 minute rounds (times 3) of cardio in the gym. Within these 5 minutes I will do a 20 second sprint and then 10 seconds of normal pace for the entire duration. This way I can replicate a match as much as possible. I also find it prevents cardio from becoming boring...
In terms of preventing reversals I would say make sure you a placing as much shoulder pressure as you can on to your opponent by having your toes digging into the ground so you can direct all of your weight into them. However, be aware of your opponent's hips and legs. If they are trying to escape I would suggest placing your arm/elbow next to their far hip to prevent movement, or, I would grab their nearest leg and drive it away from you so they pretty much end up on their side (Ryan Hall has a good demonstration of this on YouTube). However, with every situation you have to judge for yourself and if you feel they are going to escape just flow with it until you return to a dominant position (i.e. they try to reverse side control by coming up on to their side and you spin around to the other side).
Again, I am no expert so take everything I have said with a pinch of salt, however, I have experienced similar problems and these responses have worked for me.