For non-extreme hikers that want to play it safe, and are visiting India during September and early October - is it better to begin in the north-east (Sikim etc.) or north-west (Himachal-Pradesh etc.) weather-wise? Where is it warmer during the early winter?
Both Sikkim and Himachal Pradesh are going to be cold by virtue of altitude. You haven't exactly mentioned any specific places so I can't comment on which place will have colder weather, but considering typical 'places to visit' Sikkim is likelier to be the colder place (again, altitude)
Late September / early October is the end of monsoon season. So you can expect some rain in both places. You'll need to check weather reports as to what the actual conditions are. In case of roads being blocked due to heavy rains / local landslides blocking mountain roads, you're likelier to find more alternatives in Himachal Pradesh as transport links there are better (closer to populous centres in northern plains, etc. so it sees higher number of visitors).