Hello! I am currently living in Japan in a tiny one room flat. My kitchen consists of two hobs (gas), a microwave, a tiny fish grill, a toaster and I have a saucepan and two woks. For Christmas I managed to cook chicken in red wine and tomato sauce for seven, and I have cooked curries and make meatballs, so I do manage to cook decent meals in a pot, without an oven. I like baking but at the moment everything I have done has come out of the microwave. I want to make brownies and cheesecake and other cakes - I imagine I can kick-start a cake in the microwave for some minutes on a low setting, then if it is small (or cupcakes), finish in the grill to make it a little "fluffy". Is it possible to use boiling water/steam with a tight container? It is my best friend's 30th soon and he deserves an amazing chocolate cake, or perhaps a cheesecake ...
It sounds as if a great job is already being done with limited equipment, however it is unfortunately very difficult to make many cakes with a regular microwave. If you have, or can find. a convection microwave (also known as a combination microwave) then this can be used for cakes as the interior of the microwave can be preheated and the cake will cook more "normally" in the hot air circulated by a convection microwave.
There are some cake recipes that have been designed to cook in a microwave, though these are often small cakes cooked in ramekins as a dessert, and you will find some recipes on-line. It would be difficult to start a cake in a microwave and finish it under a grill as the cake is unlikely to be fully set and would probably deflate as it is transferred from microwave to grill.
We would not recommend cooking a cheesecake in a microwave as cheesecakes need constant, low heat - very contrary to the way a microwave cooks. There are some pretty good cheesecake recipes that are "no bake" recipes and we would recommend Nigella's Nutella Cheesecake on this website (see link below) and the Cherry Cheesecake recipe from Nigella Express (p81 and see link below - you could use another fruit spread, or even a fresh fruit, instead of the cherry one suggested).
Steaming is also quite gentle and cakes won't have the same lightness as those cooked in a conventional oven. It is possible to make a boiled or steamed fruitcake or a steamed sponge pudding, and again you will find recipes on-line. As a special dessert for your friend you could also try the steamed chocolate sponge from Christmas (p140). It is best served freshly made, with its indulgent chocolate sauce, though leftover slices can be re-warmed in a microwave.