Keep logging in different servers to find the one with the badge maps. To complete the New York badge, you need to do charge jumps that involve pressing down until your dude starts blinking yellow and will jump sky high when you let go of the button.
How to fly: with the words of Angry Video Game Nerd: "you press DOWN to get UP." Just hold it unreasonably long time and stand there like a dolt waiting the boosters to kick in. Yew gatta berievu!
This game is grindy as hell. To get the hard badge, you need the best skills from ALL tower research lines AND all meteor upgrades (=to kill those camo'ing thieves with ease) you can get. Skip Frenzy and Poison. The poison does close to zero damage and makes thieves go hiding, forcing you to spend mana on meteors to take them out. The gem returning skill does not help you to Brilliant grades (=returned gems have count as been touched), so skip it. The last mission: build 4 ballistas and one flame and one ice for the both entry points. Keep the towers unupgraded to maximize the exp gain. Getting the badge will take about 15 to 30 hours.
Simple strategy guide: get artillery supply upgrade, spam tanks (read: f-ing BROKEN :D ) if you can or fill the trenches with rifle or machinegun men and spam assault until the enemy loses the trench. Dump specials at enemy's manned trenches before assault teams get there.
There are not many games that do the "murder with smile" theme so well. This one is also a great game. I killed my patsies only to hear what they say in response. :)
You cannot beat the game smoothly without doing certain things and playing casual mode beforehand as the difficulty ramps up at levels 3 and 5 to the point that you have more enemies swarming you than you have bullets. Also, some random waves are simply impossible. Oh gee, spawning million ranged attackers and mosquitos when you are already swarmed. Balance issues.
SPOILER: the last challenge cave has a goblin with almost infinite health and instakill damage, to the point it can one-shot player's castle. There is challenge and then there is broken. I'm not gonna play a game longer only because dev adds a lot of zeroes to some number stats.
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I think that problem with fuel. Try to save more fuel.