Do molecules with carboxyl groups have properties of carboxyl's sub-functional-groups? We're learning about functional groups of different organic molecules, and there are some I'm confused about: carboxyl, carbonyl, and hydroxyl. I was looking online but I couldn't find a direct answer

I know that the carbonyl group is a "-CO", the hydroxyl group is a "-OH", and the carboxyl group is a "-COOH".

We learned that carbonyl groups make molecules a ketone or aldehyde, the hydroxyl group makes molecules alcohols, and the carboxyl group makes molecules carboxylic acids.

Knowing that that a carbonyl group is just a carbon double-bonded to an oxygen, and a carboxyl group is a carbonyl group bonded to a hydroxyl group, then should carboxyl molecules also have all the properties of hydroxyl and carbonyl groups (such as being an alcohol), on top of its new ones? Or, does it only have the properties associated with themselves?

Thanks.
Carboxyl groups do share some properties with hydroxy and carbonyl groups such as the ability to engage in hydrogen bonding but in terms of chemical reactivity they are very different.

The reason for this lies in the structure of the carboxyl group. Instead of the carbonyl and the hydroxyl groups being completely independent of each other, there is an orbital interaction between the three $p$ orbitals on the carbon and the two oxygens which creates a $\pi$ system across the three atoms that tends to stabilise the group, particularly when it is deprotonated. (I couldn't find any decent pictures of these $\pi$ orbitals but if anyone can please edit them in)

The most obvious property that carboxyl groups have which hydroxyl and carbonyl groups do not display (hydroxyls do slightly but nowehere near as much as carboxyls) is acidity. This high acidity is caused by the fact that the carboxylate ion which forms when carboxyl groups donate a proton is stabilised by the electron delocalisation $\pi$ system. This can also be expressed using resonance structures as shown below.



The strength of the acidity can be seen in the $pK_a$ values of carboxylic acids compared to alcohols.$$Ethanoic~acid~pK_a = 4.74$$$$Ethanol~pK_a = 16$$

Remebering that $pK_a$ has a logarithmic scale we can see that carboxylic acids are over $10^{10}$ times more acidic than their respective alcohols.

An excellent introduction to many other properties and reactions of carboxylic acids can be found here: https://www2.chemistry.msu.edu/faculty/reusch/virttxtjml/crbacid1.htm