I don't know if I've been doing something wrong, but the in-game 'Suppression' badges do not work. The first time I beat the Archmage with only water and air cards, and I expected to get the 'Fire Suppression' as well as the 'Earth Suppression' badges at the same time, and I got neither. So I thought you had to do it one at a time, so I used every element but fire, and still it did not work. The 'Big Spender Badge' states that I can't use cards that cost less than 3 mana and win, and I did that multiple times by skipping my turn. I'm not sure if the problem is actually that I can't use cards that cost 3 or less mana instead, or if skipping my turn causes me to be unable to get that badge.
There are many ways to beat Harder. I went for the farm build. Try to start off with a big farm and a hospital next to you, and a couple other farms near you. Keep two leaders, 2 builders, and rest soldiers. Take the big farm and hospital early. Don't expand too early after; try to get your survivor count higher after taking a few more farms and apartments/suburbs/motels. Then go for a church or bar. By now you should have positive income of food. Then go for laboratories etc and don't forget to throw parties when you hit 50 food. Good luck.
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Which browser are you using? Have you already tried another one or clearing your caches?