1. Do adults and children learn an L2 in the same way and? What, if any, are the differences?
From my perspective no. Children learn more unconsciously, without specific purposes, structures or anxiety. Adults, learn with intrinsic motivation but with higher fear to making mistakes, therefore less confidence, more anxiety and less fluency in most cases.
2. Can we teach a language or do people learn a language? What, if any, are the differences?
Both, to learn at same point you need somebody’s guidance or advice. The differences may lie on the context. You need to be taught a language if you are at school learning an L2, and you may learn, if you are learning English in England.
3. How is language to be learned sequenced in your current teaching context?
From basic words to more complex sentences. Also from the needs depending on the age of the students. If the students are adults it will depend on their goals.
4. What if anything can be learned by looking at past methodologies?
You can learn about the strengths and weaknesses of those methodologies, and what works for you as a teacher and for your students.