You are on a very good track.
Abhärten is mostly used when people are hardened against something by being subjected to it and increase their tolerance:
Die Spartaner schliefen in der Kälte um sich abzuhärten./The Spartans slept in the cold to harden themselves.
Your example regarding toughening oneself is also correct:
du musst dich abhärten/you have to toughen yourself
Härten is mostly used when something is made physically hard:
Die Speerspitze wurde im Feuer gehärtet/the spear tip was hardened in fire
Thetis härtete die Haut ihres Sohnes im Feuer/Thetis hardened her son‘s skin in fire
But Duden also lists intransitive use:
Beton härtet innerhalb weniger Tage/Concrete hardens within a few days time
which (I agree with comments) is unusual and use in the sense of abhärten:
sich durch Sport härten/toughen oneself through physical exercise
Verhärten is mostly used when something becomes rigid and inflexible, often with a negative connotation. Duden has good examples:
Die Not hat ihr Herz verhärtet/Hardship hardened her heart
Die Geschwulst hat sich verhärtet/The tumour hardened
Verhärtete Muskeln/tense muscles
Die Fronten haben sich verhärtet/Positions became more entrenched
But neutral connotations are also possible. Again, from Duden:
- das Feuer verhärtet den Ton
If I correctly understand your question, you are mainly unsure about whether verhärten can be used transitively. As you can see from 1. and 5., it can.