As to your ancillary questions:
What is the original Pali phrasing?
As others have pointed out, the original is:
manopubbaṅgamā dhammā, manoseṭṭhā manomayā.
manasā ce paduṭṭhena, bhāsati vā karoti vā.
tato naṃ dukkhamanveti, cakkaṃva vahato padaṃ.
Which words are able to be translated freely between mind and heart?
Some words used more or less interchangeably are citta, viññāṇa, mano, hadaya, ceto and nāma. Any one of these could ostensibly be translated as either "heart" or "mind". The only one that literally means "heart", as mentioned, is hadaya. The rest have colloquial connotations and are derived from various roots, e.g. √cit = to think, √ñā = to know, √man = to consider, or not from roots at all (hadaya = heart, nāma = mentality).
What is the overall sense of the Pali words used i.e. what are the nuances or subtleties to the original Pali?
The technical meaning of the verse is that wholesome and unwholesome deeds come from wholesome or unwholesome mind states, not mere physical acts. In this, Thanissaro's translation is misleading. The emphasis should be on the fact that it is the mind as opposed to the body, not the heart as opposed to the mind.
The origin story about the elder Cakkhupala makes this clear; during his walking meditation, many insects were killed because he was blind. The Buddha spoke this verse to show that Cakkhupala was innocent.
To get an understanding of the colloquial meaning of this verse is a bit involved, but reading the PED's article on how mano is generally used might help. Its usage here does seem to be to emphasise the mental nature of guilt and innocence. Otherwise, citta is a more common word. The word for heart, hadaya, is used when talking about emotion, e.g. hadayaṅgamā - going to one's heart (i.e. heartwarming), and doesn't seem appropriate in this context.