If you are flying into Christchurch you could do the drive over to Akaroa, it is easily a day trip, only about an hour and a half drive. It is a really picturesque little town and you can do boat trips in the harbour to go whale and dolphin watching. There are lots of nice shops and cafes there as well.
In Christchurch you should find plenty of things to do with a toddler. The botanical gardens in the city centre are easy to walk around with lots to see, fountains etc. You can take a gondola ride down the Avon River which winds through the city. I haven’t been to Christchurch since the earthquake but there is an aquarium in Cathedral Square which I think is still open.
In Queenstown you could go up the gondola and then down the luge, it is a low-speed scenic luge ride so should be fine for your 20 month old. There are lots of tourist-y things to do in Queenstown but most are adult orientated, adventure-tourism type things. There are a range of cruises you can do on the lake, or you could drive over to Wanaka for a cruise from there. There are a few good day or half day trips to take from Queenstown – Arrowtown is an old colonial town and good for a look around. Try Patagonia (in Arrowtown and in Central Queenstown) for amazing chocolates and desserts!
In regards to the cold it depends what you are used to but it won’t necessarily be too bad. In Christchurch it will often be around 10-15 degrees throughout the day.
I haven’t been myself but if you were up to the drives Nelson and Kaikoura would also both be great places to visit. Nelson is the sunniest place in NZ so would be a nice escape from winter