1) It's my understanding that a sponge and a starter are the same thing except for the hydration %. For example, if you took some starter and mixed in flour so it was somewhat dry, it would produce a sponge. However, if you took that same sponge and let it sit in the fridge for a long time, you would be back to a "runny" consistency starter as the food source was consumed by the yeast. I am not a expert on the terminology so correct me if I am wrong. The reason most recipes call for a "sponge" is because you took the time to feed the yeast and make sure it was actively growing well prior to putting it in your dough and made sure it wasn't in a "dormant" state. Once in the fridge for a long time, the yeast will enter into a lag growth phase and not be very active. That is why you typically don't use a starter right from the fridge into your dough, because you will get less activity out of it.
2) Use whatever container you can get your hands on, I don't know about metal. Something to be mindful of is contamination. Some cultures are fairly fragile ecosystems and they are easily contaminated with other strains. If you are growing more than one culture, be very clean in your technique so you don't contaminate one culture with another. (I would use a separate utensil for each.) that sort of thing. If you use small amounts of starter while growing in the container, don't overfeed them. You need to let the culture grow to gain size, before you add more flour to it. (Another reason for making a sponge.) Be sure to store some of your culture in a small little vial which has never been fed. This way if you contaminate a starter, you can start over and innoculate again. I have read of some people dehydrating a sheet of starter so they can permanently store it. I have never tried this but it seems like the viability would be low. In the lab, we store cultures @ -80C with 20% glycerol in the cells so they don't pop from freezing.
3) Even in the fridge, the culture will grow, albeit VERY slowly. I would try your seedling mat to get some warmth there. If that doesn't work, try a heating pad on low. I believe this is more critical when you actually go to make the bread or the pizza. If you don't get a proper rise before the dough dries out, you will be left with a tortilla! I would also cover it to keep the humidity in there. You might have to proof your sponges longer than normal if the weather is really cold. However for the actual dough rise, you need to find a warm solution.