If your doctor advised you work modify hours or part time, are you able to apply for Ei sick leave or eventually Ltd while you work part time?

You need to check with EI about this. Regarding LTD, some policies allow for income to be earned up to a certain amount while receiving LTD. However by working while applying or receiving LTD the insurer may take the position that you are not in fact disabled and not pay you at all. Check the wording of your specific LTD policy.