Years ago I was working a summer construction job when I was in college, and one of the regular employees swore that the hotter it was, the more hot coffee you should drink, using this very reasoning. I thought to myself (since I had already been branded the Dumb College Kid and didn’t need any more guff), “You must be a lizard.”
Regardless of outside temperature, consuming anything cold reduces your core temperature, and blood circulation to/from your extremities will cool your whole body. Hot stuff does the opposite. Would you give ice water to a hypothermic person, or hot water to a heat stroke victim?
I would certainly give hot water to a heat stroke victim, if I didn’t have any cold liquids available. When it’s hot out, it’s very, very important to drink plenty of water, and while cold water will be a little better than hot water at cooling you off, it’s a comparatively small difference.
As far as I’m concerned, ice cream IS a liquid. Frozen, yes, but it’s in a liquid form before it hits your stomach, that’s for sure. I don’t know about slowing digestion, but something full of sugars and fats does need to digest, as opposed to something like water.
Speeding up metabolism should make you feel warmer, since the body produces waste heat as it does work.
I don’t think the temperature of liquids has any major effect on metabolism - certainly we can’t say without knowing the surrounding temperature and whether your body is already working to stay warm or trying to cool down.