You're making a big assumption that there is a significant heat source transferring heat to the water other than through the air. Otherwise in both cases you still rely on heat transfer from the air, it's just that in one case it has to heat up the water first. In a typical experimental setup this might be from a bowl sat on a countertop at ambient temperature or something, but if the bowl had a very low thermal mass and was supported by very thin members then the draining setup could easily melt the ice quicker.

Not really. When you get out of a hot shower and move into another room you feel cold because there's a lot of water evaporating off your skin, taking latent heat with it. Also, if you have ever been very cold you will know that just sitting in front of a fire doesn't immediately make you feel warm; even though you are warming up straight away it takes time for you to feel warm.

100% this. Heat transfer from the foot itself is definitely a factor, but I don't think it's the biggest one. While by no means sealed, the air within the blanket becomes pretty stagnant and holds heat while you're underneath it. When you stick your foot out you are creating a significant opening as well as air currents and turbulence from the movement that allows a large exchange of warm air under the blanket with cold air from the room.
You can easily test this by very gently poking your foot out of the covers while keeping them tightly wrapped around your ankle. You don't get the same cooling effect (I just tried it while I was writing this). P.s. if I'm hot in bed I just kick my leg up high, which lifts the covers and lets all the warm air out/cold air in. Don't need to leave my foot stuck out to be quickly cooled down.