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.
Which browser are you using? Have you already tried another one or clearing your caches?