I received quite a few cuttings in past few weeks from generous members here and the other forum. I understand rooting cuttings at this time of year is a some sort of challenge, particularly for a newbie. But I just can't wait, hoping they grow to decent size and start to take off after weather gets warm.

My family don't like the house too warm so the thermostat is set to 62F while away and 68F after returned. There is a south facing window that is the most sunny spot in the house. This is where I am planning to grow cuttings after they have green growth but I would imagine it is couples of degree colder than where the thermostat located because it's just two layers of glasses from outside. Given long raining season here in Vancouver, it wouldn't be heated up by sun very often.

First question, is the temperature high-50 to high-60 enough for rooting cuttings and keep them growing well? Is a heat mat needed or useful in this case?

I found the warmest spot in my house is furnace room which also has hot water tank there. The temperature in the room is between 70F to mid or high 80F but fluctuates a lot depending whether the furnace or hot water tank is fired. Is it okay if I moved cuttings into the room after sun goes down? That said, the cuttings have bright but colder in day time but dark and warmer at night. Will they benefit from higher temperature at night?

Any advice is truly appreciated.

70's to 80's fluctuations are just fine. I'd put them by the furnace at first.

You have to be careful with higher night temperatures.   https://en.wikipedia.org/wiki/DIF_(technique)   As Lewi said, they should be fine in the furnace room until rooted, but I wouldn't move them in and out.  And be aware that the higher temperatures might produce leaves before roots.