The temperature of the ingredients is very important when making cake. You will have problems if you start from cold ingredients. I'm afraid that there are no good solutions if you don't have a warm room, all work arounds are quite laborious.
Microwaving the butter is not a good solution. It is a very uneven method of heating. You always end up with too-hard spots and melted spots beside the somewhat softened butter.
The proper way to have butter at the correct consistency is to keep it for at least overnight in an environment which is between 20 and 30 degrees Celsius, and has ambient heat. Putting it on top of a working oven and similar is problematic. If you don't have a room in your house which is at that temperature, you might consider making a water bath for your butter. Warm a Dutch oven with water in the stove until it reaches 30 Celsius, pack the butter in a smaller container, float it in the Dutch oven. Pack the whole thing in a few blankets. It should soften the butter overnight without getting too cold (you said your room doesn't fall below 10 degrees).
A workaround is to cut up the butter into very thin pieces (2-3 mm) and then put cream them with the sugar more than usual. The friction from the mixer will warm it up. This can work well if you have a stand mixer, but if you are doing it with a hand mixer, you'll need to stand there for 15-20 minutes just beating the butter. A water bath might help, but you really have to pay lots of attention not to melt the butter.
You can also consider replacing the butter. Shortening and margarine are much more beatable at low temperatures than butter. You'll get worse taste though, and some change in texture. You have to use a solid fat, not an oil. If you use a margarine, use a full-fat one, not reduced-fat spreads.
The butter is your most obvious problem, but you'll never get perfect results if your eggs are cold. The texture might be good enough for you when you bake with cold eggs; I've seen many bakers who don't mind it. But if you are really picky about your cake quality, you also need to warm the eggs overnight. Alternatively, you can quickly warm them in a water bath after breaking.
Adding milk or other liquids is counterproductive. It will change the texture of the finished cake a lot. Your recipe is a slightly modified pound cake, standard pound cake recipes don't have liquid (and neither do many other cake types). The only reason for dairy I can imagine there would be to add some acid for the baking soda to work - but then it should be buttermilk or yogurt, not milk. The more liquid you add, the less cakelike and more quickbreadlike your result will be.