1. Warming cold noodles in a hot pot of soup?
2, reheating boiled chicken on shabbos, is there a way?
3, dipping challah in hot soup?
4, heating a dessert on hot plate where margarine will melt?
Although we can't heat a food to the point of the heat where the hand would not touch it as its so hot, we can reheat foods under certain circumstances. This is because a food can't undergo halachik cooking twice. 
If you recall cooking is defined as the changing of a food or substance by use of heat. 
However in the case of a previously cooked food this doesn't hold true as the food has already undergone a major change when it was first cooked, so reheating isn't considered cooking.
There is a big difference between solids and liquids.
To be able to reheat a food, it must be a solid food and dry.
A cooked liquid can only be reheated if still warm. 
Additionally a food may only be reheated with the same method that was used to heat and prepare it the first time, eg a roasted food by roasti
ng it and a cooked food by re cooking it.
Some examples,

1) it's allowed to take a cold baked potato from the fridge and to place it near flame where it can become hot.

2) one can place cold boiled chicken from the fridge into a hot pot of soup (1st vessel) that was removed from the flame ).
3) matzah or challah may not be put into a hot pot of soup since challah and matzah are baked, and you are now cooking them. 
How dry does dry have to be?
It doesn't have to be literally dry. As long as no significant accumulation of gravy or liquid its OK. Eg meatballs are considered dry even if moist, as long as not in gravy.

Margarine, congealed fat are regarded as "dry" as there current status determines them and they are dry at room temperature.
Examples include rewarming chicken with congealed fat or a dessert being rewarmed that has margarine in it. 

Anything that is a liquid can't be rewarmed. 
Examples include meatballs in sauce, pouring ketchup into a hot pot of chullent, a pot of soup etc. 

The reason liquids are different is that heat doesn't have an intrinsic transition in a liquid, which therefore means that a change in temperature of a liquid to a hot point is itself the state of cooking.

As long as the liquid remains warm it can still be reheated. "Warm" would mean it could be used for a hot drink. Eg if leftover soup is hot enough it would still be enjoyed, it can be poured back into a the hot pot.

It should be noted that only intense dry heat radisting from a direct flame is an issue, therefore boiled chicken or noodles can be put on a hotplate isn't intense enough to cook it. 

Foods that were both roasted and cooked go after status of last process, eg a lokshen kugel.

So back to our questions.
1) Warming cold lockshen in hot soup ok as long as pot off the fire, as its same way the lokshen was cooked.
2) Reheating boiled cold chicken is ok either by placing it into a pot of soup or by putting it onto a hot plate as the heat isn't hot enough to be an issue.
3) Dipping challah into hot soup is an issue as the challah was baked and you would now be cooking it. The way to do this permissibly would be if either the soup in bowl is no longer terribly hot or if your the hot soup is in a 3rd vessel.
4) Heating a dessert is ok even though the margarine will melt as its considered a dry food as we look at it's current status.

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