You must use a live yoghurt if you're going to use commercial yoghurt as a starter culture. Look for plain yoghurt only. Non plain yoghurts tend to be very high in surgar which may impact the culture growth.
There are different strains of yoghurt culture out there. Look for words like 'pro-biotic' and 'pot-set' and of course, 'live culture.'
Greek style yoghurt is usually live, but it a high acidity yoghurt which is not for everyone. Greek yoghurt is traditionally drained of some of its whey. The whey can be used too.
Yoghurts which say 'pot set' on the label usually contain live culture (make sure there's no gelatin listed on the label). Not all 'pot set' cultures are generational. So, the yoghurt may work well first time and become weaker in each subsequent generation.
Temperatures between 105 and 109 are good for warm culture yoghurt. There are room temperature cultures as well, but I've not come across them in regular commercial use. Health food shops may have starter cultures available.
Using a full fat milk will give a firmer set. I'd suggest 8 - 12 hours for good culture establishment. The longer the culture is left to breed the better the yoghurt set will be, but the higher the acidity in the end product.
No need for the ice bath. Just put it straight in the fridge. Suggest at least 18 hours set time before consuming.
When using your yoghurt, always leave 2 tablespoons of the current batch to start the next generation.