"However, it is cold today."
でも is a little stronger. Think of it as で and も。で as in accepting the previous statement as truth, も as in 'also'. So it means "it's like that, but also ...." I find しかし as more hesitant. If i were to say i don't like to cook but i make my own dinner every night i would use しかし。If i were to say i don't like Japanese food but I love ramen I would use でも because the second sentence goes more against what i said in the first.
つめたい is used for like food and stuff. つめたいコーヒーが好きです。You wouldn’t use つめたい with weather. You use さむい or すずしい.
No, that sentence means more like "but, it's cold today." And they were looking for "however," which is really similar, but still different in connotation.