Hi Chin family,
A couple of suggestions:
• 11 nights altogether in Sydney is a long time for a visit, even for children. While I understand that with 4 young children and an elderly parent you don't want to go flying around the country(!), there are some lovely spots that are not too far away from Sydney that you could spend some time in to see a little bit more of NSW. I'd leave your 7 days for Sydney and consider spending the other four somewhere else.
• You've indicated that you want to hire a people mover for your trip, but much of that you want to spend in the city itself. Be aware that the Sydney CBD is not great for driving to during the week, and parking is horrific on weekdays (especially in terms of price). For that reason, you might want to investigate staying in Sydney's suburbs somewhere in some serviced apartments that are close to public transport and which can accommodate you all, rather than the outskirts of the city and trying to drive in somewhere. I'd stay near a train station that has a lift. That way you can use your car when you need to and leave it at home when you don't.
• Perhaps consider the Hunter Valley or South Coast or Southern Highlands for a few nights as an extra place to go to - they are all within 2-3 hrs drive of Sydney and all very nice areas. Some areas have more than others for children. See www.visitnsw.com/things-to-do/family-holidays for some suggestions on family-oriented attractions.
• There aren't that many farm stays close to Sydney but there are a lot more in the areas just beyond Greater Sydney - see www.visitnsw.com/accommodation-farm-stays for some suggestions.
Hope that is helpful - feel free to ask more questions.