Firstly, сотейник spells with an 'o' after 'c'. Secondly, as a native Russian who has lived in Russia all his life I can assure you that this word is rarely used these days. More often people call it глубокая сковорода or кастрюля с длинной ручкой, depending on how deep it is. A saucepan is usually called кастрюля when it's depth is bigger than its radius. Кастрюля does not necessarily have a long handle. It may have two symmetrical small handles instead. I guess, in that case, 'pot' would be a more appropriate word for the utensil. Кастрюля made of glass and designed for cooking in the oven is called 'casserole'.

Hob = hotplate. Пилта is the whole cooker/stove. There is a semantic difference although the meaning is understood. Interestingly, Yandex Translate gives пилта as an alternative translation for "hotplate", but I don't think it's fair to say a hob and a cooker/stove are the same things for the purpose of this exercise.

Yeah, yeah, I get what it is, I just never heard the term either. Is it British or Australian or Scottish or what? Here in the states it's called the "element".

Where is that term used?

I did: eyeballing it, out of the 100 or so, three or four were кастрюли -- not very convincing. The rest were the almost flat rectangular baking vessels from glass, ceramics and metals that go into the oven to make casseroles and lasagnas. With handles and without.

Форма для тушения/выпекания -- sure, that's exactly it. I mostly agree about противень (baking sheet) and pies. Some exceptions though would be the cases where you need to make something that takes liquid or easily flowing dough, such as бисквит, шарлотка and so on: you would use a casserole for those. I actually make a cabbage pie in a medium sized casserole rather than on a baking sheet, it's just easier to handle.

I appreciate the etymological details you frequently include in your posts, by the way. Not being a linguist, I find them very interesting and some are totally unexpected.