1. No. It all depends on the person and the context. I use both 'shall' and 'whom' whenever appropriate.
For example, in conversational British English, it is quite normal to make a suggestion by saying 'Shall we go for a drink?'. Note that this is not formal language at all. It comes out as 'Sh'we go for a drink?', and it is perfectly natural, relaxed everyday speech.
'Whom' is a useful way of making it clear that someone is the object of a sentence. It's optional as a direct object, but obligatory as an indirect object following a preposition. For example 'The person for whom I felt most sorry was the grandmother.' This usage is not at all uncommon.
2. Yes. They always sound very awkward and stilted. 'Who was this picture painted by?' is the natural modern-day construction. The only people who would say 'By whom....?' nowadays are native speakers trying to be deliberately formal (often as a joke) and non-native speakers who are translating word for word from a parallel construction in their own language.