"what provides wholesome pleasure and joy in a context of mental states and Buddhism?"
In general, anything that weakens greed, aversion and delusion is a potential source of wholesome pleasure.
From the suttas....
Abandoning what is unwholesome is a source of pleasure (AN 2.19):
because this abandoning of what is unskillful is conducive to benefit and pleasure, I say to you, ‘Abandon what is unskillful.’
Similarly, the practice of sīla (the virtue group of the noble eightfold path) incurs in wholesome pleasures (AN 10.1):
“Ānanda, the purpose and benefit of wholesome virtuous behavior is non-regret.”
“And what, Bhante, is the purpose and benefit of non-regret?”
“The purpose and benefit of non-regret is joy.”
...which continues in a chain leading to jhāna and, further on, liberation. In the context of meditation, jhāna is specifically said to be a superior pleasure compared to sensual pleasures (SN 36.19) and, for example, seclusion and samadhi are sources of pleasure (in the first and second jhānas).
Also, the following are benefits associated with wholesome pleasure obtained when metta practice "has been pursued, developed, and cultivated, made a vehicle and basis, carried out, consolidated, and properly undertaken" (AN 8.1):
(1) “One sleeps well; (2) one awakens happily; (3) one does not have bad dreams; (4) one is pleasing to human beings; (5) one is pleasing to spirits; (6) deities protect one; (7) fire, poison, and weapons do not injure one; and (8) if one does not penetrate further, one moves on to the brahmā world.
Particularly, the abandoning of sensual pleasures (unwholesome) in favor of exclusive wholesome pleasures is taught in accordance with one's commitment. In this case, for those who enjoy sensual pleasures, the Buddha taught four kinds of happiness that may be achieved: "The happiness of ownership, the happiness of enjoyment, the happiness of freedom from debt, and the happiness of blamelessness" (AN 4.62).
Also, the Buddha said that "Longing and unrighteous greed are a defilement of the mind" and taught them the proper way of gaining wealth, fame, long life, and a happy destination after the break up of the body (AN 4.61).
Thus, the preservation of the precepts, the practice of sīla, and teachings such as the ones above are the proper ways (i.e. according to the Dhamma) to engage in "an activity one enjoys which isn't harmful" and "fulfilling ambitions linked to one's aspirations".
"And, again from the perspective of mental states, is happiness always reliant on subjectivity to a degree? Here, I mean to ask whether one's mood, or personal interest, or one's personality always play a part in generating pleasure and joy (aside, perhaps, from in meditation)?"
Sure. For example, while a certain person may find herself delighting in, say, virtuous behavior, someone else may not. Some people may delight in concord while another may delight in discord (MN 41). Finally, some people find peculiar things as source of pleasure:
“What, bhikkhus, is the way of undertaking things that is pleasant now and ripens in the future as pain? Here, bhikkhus, someone in pleasure and joy kills living beings, and he experiences pleasure and joy that have killing of living beings as condition. In pleasure and joy he takes what is not given… holds wrong view, and he experiences pleasure and joy that have wrong view as condition. On the dissolution of the body, after death, he reappears in a state of deprivation, in an unhappy destination, in perdition, even in hell. This is called the way of undertaking things that is pleasant now and ripens in the future as pain.
-- MN 46