Is one simple teapot – Allowed to be used for all three settings. For instance, for breakfast (dairy), Parve (keep’n an isolation from dairy and meat) Dinner (meat). May one teapot be of use for all three settings or a person must obtain three different kinds of teapots? Alternatively, cups must be changed but the teapot on its own (I’m lost about that part).
Yes it would be halachically considered writing. The reason being, that the picture that is made on it will remain that way, at least temporarily, which would be kosaiv m’drabonon. Therefore to write letters, a picture or even a shape that has meaning to it would not be permitted on Shabbos or Yom Tov. Regarding making squiggles etc. that are meaningless, technically it would be permitted, but it is not recommended, because one might inadvertently write something that has meaning to it, therefore it is best not to play with it on shabbos.
M:B 340- 20, 25(1), Biur Haacha D:H B’mashkin, Zachor Vshamor (Kosaiv) pg. 4, R’ Y. Cahen shlit”a.