Diplomatic immunity means that diplomats are immune from prosecution in their host countries. It does not necessarily mean that they can be sued or prosecuted in their home country. Whether the diplomat is subject to the home country's laws and courts is up to the home country.
In most cases, it seems, diplomats can escape prosecution altogether by reason of their immunity. If the home country wishes to dissociate itself from the diplomat's actions, it can waive diplomatic immunity to enable prosecution in the host country's courts.
There are some interesting examples, some of which touch on your question, in the Wikipedia article on diplomatic immunity.
With respect to your example of an apartment lease, it appears that a diplomat might not actually be immune, since the Vienna Convention on Diplomatic Relations of 1961 provides for some exceptions to immunity, including
A real action relating to private immovable property situated in the territory of the receiving State, unless he holds it on behalf of the sending State for the purposes of the mission
Interestingly, the Wikipedia article lists two prosecutions under home-country legal systems that both involve Romania, and both involve fatal accidents in which the driver had been drinking alcohol. (In one case, Romania was the sending state, and in the other, it was the receiving state.)
In one case, the driver was an American Marine assigned to guard the US embassy. He was tried in a court martial, presumably under the US military code, which no doubt penalizes drunk driving, but that's just a guess. The court may well have applied Romanian law.
In the other case, Romania apparently prosecuted its chargé d'affaires in Singapore after a fatal automobile accident. It's not clear whether he was charged under the Romanian or Singaporean law.