Thermoelectric generators (TEGs) are heat engines. They indeed "use up heat" to produce useful work. What happens is that some of the heat that passes from the hot reservoir through the TEG and to the cold reservoir is converted to useful work. The typical efficiency of a TEG is about 5%, but that number depends on the temperature difference between the hot and cold side, so for small $\Delta T$ (around 1 K for example), an efficiency below 1% is not uncommon.
Some of this useful work is lost through Joule heating inside the TEG itself (and to the load to which we can assume it is connected), enhanced or hindered by the Thomson effect and possibly other thermoelectric effects if the TEG contains anisotropic and/or inhomogeneous materials.
In these kinds of analysis, we assume that the hot and cold reservoirs are infinite and that their temperature do not change. So the temperature isn't lowering at the hot side even if the heat engine is producing a useful work (but you're right, it would if the reservoir wasn't "big" compared to the heat engine itself). However the temperature of the heat engine itself may either increase (if the Joule effect dominates for example) or decrease compared to if it wasn't connected to a load, i.e. if it wasn't performing useful work. Indeed, if the Thomson effect dominates over the Joule effect (which is possible for many materials under certain conditions), the TEG can be colder when it is powering a load compared to when it isn't producing any useful work.
This is the realm of non equilibrium thermodynamics, where there are entropy fluxes and where we usually apply Onsager reciprocity relations (we assume small perturbations that drive the system out of equilibrium). Everything is consistent with thermodynamics, no fundamental law is violated.
To answer your other questions, yes kind of. The molecular vibrations could be replaced by phonons to make the questions a little bit more formal. The answer is yes, it is possible to extract energy out of the phonons (or thermal atomistic excitations). As soon as a temperature difference exists across a thermoelectric material (such as a TEG leg), a potential difference builds up. If the TEG is connected to a load, it produces a current and thus a useful work.
Now, as to why an electric potential builds up when a temperature difference is established across a TE element, I suggest you look up the Seebeck effect. A quick (and dirty) explanation is that the electrons are affected by temperature and that they tend to diffuse from either cold to hot or vice versa, thereby creating an electric field and so a potential difference which can be used to power up an electrical circuit.