Thats a lot of questions. I'll try and give some insight into them...
1. The heat energy from the water will warm your skin but only fractionally. The warm water stimulates the nerves under your skin that detect the heat and pass that information to your brain, but they need very little energy transfer to trigger. Most of the heat energy will be lost into the air, but still only a small amount, if you collect water at the shower head, and then collect water at the shower floor, the difference in temperature will be quite small. Back to you feeling warm, the body detecting the heat moves blood to the outer surface of the body to make sure the body stays at a constant temperature, this is why you look red after a hot shower, or why a burn is red and feels hot, it's the blood not the applied heat.
2. Breathing air in a hot wet environment feels harder mainly due to the water vapour content. The air is denser and your lungs need to expel as much of the water vapour as it can as well. You get the same effect in a sauna or jungle.
3. You feel cold due to water evaporation. Evaporation is how your body naturally stays cool, i.e. sweating, your body releases moisture this moisture is warmed by your body and then taken away by the air, so your body has lost the heat required to warm the water. Any air movement across your skin will evaporate moisture, even the air movement caused by you walking. In the shower the air is saturated with water vapour so evaporation is small, if you move out of the shower the air is not saturated so evaporation happens faster and you feel colder. Go outside in a gale and you will get really cold (and arrested if you forgot to get dressed) even on a sunny day. Remember from answer 1 that the blood rushed to the surface of your skin, well now some of the heat lost through evaporation is taken from the blood so this cooler blood is passed back to the body core, your body detects this change in temperature and triggers an over cold reaction, your body needs to move so that heat generated in the muscles can warm the body, you will get involuntary muscle spasms or shiver.