I don't live in that particular city or know anything in particular about that specific programme or university, but I do know about the general costs although they vary a lot and may be slightly different in that part of the country since I live in Copenhagen which is more expensive but I'll try to give some insight anyway.
There is no set minimum wage, it's negotiated between the employer and the union that they cooperate with. My estimate would be around 17-20$ per hour for absolute minimum wage. Food is varies a lot, but junk food is probably somewhere between 5-15$ for a meal (like a wrap with meat, a sandwich or a standard McDonalds meal would be somewhere in there). If you cook yourself, as long as you're not planning to make gourmet food or whatever, it shouldn't be all that expensive.
I'm not particularly familiar with the job situation over there, so I don't really know if you'd be able to get one, but I know that all Danish students over 18 years receive a set amount of money each month depending on the income of their parents (though I have NO IDEA if this applies to people without Danish citizenship, but I'd imagine that it doesn't). Most schools provide the books for their students (at least at a high school level where I'm currently at, though I don't know if that applies to universities). If you had to pay for books yourself, it could be anything from 200-500$ I guess. I'm sorry I don't have the exact information but that's a general estimate I'd make.
If you just need a place to sleep, you could probably get a really cheap place meant for students that should be around 200-300$ per month (again I'm not sure if this only applies for danish students) but a standard apartment will probably be around 500-800$+ per month (depending on location). I really can't tell you how easy it is to get a place to live though, but it seems like quite a few people are selling 1-room apartments (just did a quick Google search)
To my knowledge there are no laws regarding open grills or cooking in the open as long as you're not cooking for 50 people every day. Personal use and a party once in a while should be absolutely no problem as far as I can tell.
I hope that provides some knowledge at least, and I apologize that I can't be more specific but I dont live in Kolding and am unsure how different it is from the conditions in Copenhagen.