What are the keys to earn Gold on Opening Gambit?

Solution 1:

You don't need much base defense. Try getting by without building any bunkers, and just defending early with the marines that you build, and later with tanks. You'll definitely need a good wall-off and units correctly positioned on the cliff to do damage to the zerg as the come around the corner. You can afford to expand pretty late (you only need it for the gas), once you have the majority of your tanks sieged on your cliff. Expanding late will allow you to not have to spend any extra money on base defense.

You should not stop producing SCVs for a very long time. It should be at least 10 minutes where your command center is NEVER idle. Get an orbital command as soon as your first barracks finishes, and use MULEs every time its up. Pump marines non-stop from your first barracks (no add-on). As gas allows, get your factory up and pump tanks until you hit 8. Get siege when you have enough gas to research it without stopping tank production. As your 1-base economy starts to peak, you will have enough extra minerals that you can't spend with your limited number of production buildings. At this point, build a second barracks and you'll soon be at 30 marines. Once at 30, put tech labs on your barracks and get ghosts as gas allows. You should also have extra minerals, since 2-rax marine, SCVs, and a factory should not be spending it all. When you have 400, build a command center inside your base (you'll float it down when its done). Move SCVs down from your main, and prioritize gas. You should be close to 8 tanks by now, so put all your gas into ghosts. If you can't spend all your gas on 2 rax ghosts, get more.

I got gold on this challenge on my first try. It shouldn't be too difficult if you follow good macro fundamentals (constant production, don't supply block yourself, correctly prioritize your resources).

Solution 2:

Your strategy should be ok, but remember that you can switch the addons by lifting up the buildings, that should allow you to speed your game up.

Start with 2 Barracks with reactors and pump marines while you get the factories up and running. As soon as the marine count reaches the goal, lift up your barracks and place them somewhere else with tech addon and pump ghosts, if you have the minerals then add a third barrack.

I didn't wall off (i never do, not even in ranked games) and never built a bunker, you should be ok if you have your marines up at the ledge where they can shoot down at the attackers without any shooting back.

If i remember correctly my expansion was up around 20 supply and i lured the zerglings that attacked shortly after to the ramp by sending a marine to them and retreating to the other marines that finished them off quickly. After that your marines on the ledge together with the tanks will kill everything coming in before they reache the expansion, as long as you stay up there.