We are a family of 3 with a child at 4 years old. We have 2 days vacation in Hong kong.

As for your questions, here are my suggestions:
1. Airport shuttle bus no. A21 can take you from the airport to Middle Road, Nathan Road. Upon getting off, you can walk to your hotel in a few minutes. The bus costs 33HKD per person and takes around 65min. For children between 3 and 12 years old, they can take the bus with 50% discount.
2. Your hotel is very close to the Tsim Sha Tsui MTR Station, so you can go there and take subway Tsuen Wan Line to Lai King. Then you need transfer to Tung Chung Line to Sunny Bay, where you can easily take the Disneyland Resort Line to the attraction. The whole journey will cost you around 17.9HKD and take 40min.
3. Temple Street is a long one. Where would you head for exactly? Anyway, to No. 1 Temple Street, you can take bus no. R11 from Disneyland and get off at Man Ming Lane, Nathan Road. Then you can walk to the street. For the journey, around 38HKD per person is needed and it takes about 30min or longer.
4. From your hotel to the peak, you can take Tsuen Wan Line at Tsim Sha Tsui Station and get off at Central. Walk to Garden Road, Central, and you can take the peak tram to the peak directly. You need pay for around 17.8HKD and around 50min is needed.

5. From the peak to TST Promenade, you can take the same transport as question 4( peak tram to Central and then Tsuen Wan line to TST. Afterward, you can take a taxi to your destination in several minutes). The taxi fare may be 20HKD.
6. Back to your hotel, it is suggested to take a taxi, which can take you to your hotel within a few minutes.
7. You can still take bus no. A21 back to the airport at that time.

Thank you so much for your reply Danielle. Just one last question..we have 3 large baggages weighing almost 20 kgs plus 2 handcarry bags. Taking the bus is great. We are also wondering if we take a taxi from the airport to the Day and Night hotel, which taxi color should we take? How long is the travel time? How much will it cost?..we will then decide what to take, bus or taxi after a reply..thank you so much again in advance.

If you have that kind of luggage, I think the taxi will be much better.
The taxi fee is around 230 HK$ and you need to take a red taxi. The time is around 40 to 50 mins.
Hope it is useful for you.