1. Yes, you can get EI maternity benefits. They will pay you 55% of your wages for up to 50 weeks You should visit Service Canada to ask about the details. What you would need is a record of employment from your employer which states how many hours you have worked and how much money you have paid into EI etc.
2. Filing through your spouse would be faster. Otherwise you are taking time away from your work because of the baby and it will not be counted as a part of the 24 months you have to work total.
3. Your agency is probably right. Your work permit is based on being a live-in caregiver. If you move out of the house, you are not a live-in caregiver anymore. You can call CIC to ask about this, their number is at
http://www.cic.gc.ca/english/contacts/call.asp