We have had to live like you with no kitchen to use and it is so hard, but with an electric skillet, microwave, toaster oven, hotplate, waffle maker/griddle and crockpot, we managed quite well. (this was before we knew how icky the non-stick and microwave were!--however you use what you have to in situations like this)
We didn't have a fridge, but did have a freezer, so we kept a lot in the freezer. We used a shelf in my mom's fridge for some items. (We also didn't have runny water or a bathroom in our apartment! So we carried what we needed from my mom's place.) We did have WIC and food stamps at the time (yes times were hard!). I told them simply that we rented an apartment from my mother. We had separate cooking space and I kept my food separate from all the other food in the house. This was all that was required. Being able to keep *your* food separate. So that is something to keep in mind. Heck, maybe you can find a small fridge on Freecycle and put it in your apartment! Don't be afraid to ask for items on freecycle.org too! You would be surprised what you can come up with. Craigslist is good too, but there is usually a cost. Freecycle is free.
As for cooking. Get the best ingredients you can get. You will save money by avoiding soda pop and all other junk--processed foods anyway! Make each meal the best you can nutritionally. Most bang for your buck so to speak. Eggs are such a perfect food and can be eaten in so many ways. Get as many of them in as you can.
Buy organic foods in places where they are most dangerous eating them non-organic. Use oatmeal and other hot cereals for breakfast--cold cereals cost too much money and nutritionally void. Whole grain pancakes and waffles, biscuits, etc are always a good hit. Make puddings, make your own flat breads like tortillas--add some good cheese and you have a nice lunch with a healthy soup and it is even better.
There are a lot of ways to make simple easy meals using what you have cooking wise. You just have to be creative. Taking some time to really research things and make a plan will go a long way. You have made the first step by asking for help and ideas here.
Making a weekly meal plan is a good idea too, so you shop all at once and of course don't go when you are hungry. Avoid convenience foods. Think cooking from scratch. It is a bit of work with your situation, but worth it.
We live on a tight budget here now. DH is disabled and a house full of youngins.
I can totally relate to doing it on a tight budget. Once we gave up all convenience foods and make it all from scratch, we need less and it takes less money. We raise our own foods, eat all organic cooking like they used to eat before all the "convenience" came to be.
I know this isn't possible where you are right now, but it is something to think about for the future.
I also wonder if there is any way some mamas here could help by sharing with you items we don't use anymore? Just a thought. Especially with anyone that might live near you.