Do molecules with carboxyl groups have properties of carboxyl's sub-functional-groups?
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