Relative stability of carboxylic acid derivatives
Firstly "stabilised" is very ambiguous. "Stable" is always defined with respect to something. The proper phrasing here should probably be stable towards nucleophilic attack.

For the anhydride - yes the actual structure is a hybrid. That just means that both C=O have to "share" the central oxygen's lone pair - in a sense each of them get half resonance each - which means that both C=O's are not very stabilised.

I don't buy the "less likely" explanation. Your teacher is trying to say that there's a "50% chance" that at any one point in time a particular C=O is stabilised, but resonance isn't something that you flip a coin and decide which way it goes.

For the acyl chloride, it is really to do with the leaving group ability of Cl-. See this question for a discussion: Do acyl fluorides exhibit resonance donation or inductive withdrawal?

Essentially the idea is that since $\ce{Cl-}$ is a better leaving group than $\ce{OMe-}$, for the acyl chloride (X = Cl), resonance form 2 contributes more to the actual structure than it does in the ester (X = OR):



As you can see, resonance form 2 makes the C=O more susceptible to nucleophilic attack because of the positive charge associated with the carbonyl group. Therefore, any substituent X that favours resonance form 2 will make the carbonyl group less stable towards nucleophiles.