I would like to share 2 big tips:
1. The more obvious one: Remove the wallet password, you can put it back when you're done playing.
2. Use the debug window. This eliminates the delay in having to click the "are you sure" window after sending.
A quick guide on debug window:
Go to the help menu, then debug window, then click on the console tab. You'll see a window where you can enter commands at the bottom. Type "getinfo" into it, just as practice. See how it works? Good. To send to an address, test it on yourself, the command is:
sendtoaddress <address> <amount>
it automatically adds the miner fee on top of that amount (usually only 1 doge), so make sure to send a tiny bit less than the entire contents of your wallet. This is an important point since its horrifying when it says "insufficient funds" and you lose precious seconds because of that.
So when it's GO time, have the command all typed out with the amount, then when the address appears on the website, copy it, paste into the right spot, slam the enter key, win.
edit: I see the site owner liked this and is spreading it in places. I didn't want to highlight the fact that I've never lost a round, I feel like a jerk admitting that part, but its true. I've been chatting with him in PMs and told him that for some reason I'm always in the top 3, and I am not cheating. I finally came to the conclusion it must be the debug window thing so I decided to share. I think if more people win more often, it will get popular, since a dead game would kill the free lunches I've been getting from this. And I'm not going to just "play slower", that's stupid.