I'm specifically thinking of the cult of saints, which resembles the notion of demi-gods; the cult of Mary which resembles the cult of Isis (Stella Maris, etc); the consumption of the blood and body of Christ transubstantiated as bread and wine, which resembles the similar Mithraic ritual consumption implied in the Tauroctony; the connection or lack of connection between Christianity and the other fashionable death-and-resurrection mystery cults of the East; the Devil as Pan interpretation; Augustine of Hippo's obsession with Neoplatonic and Manichean thought; the use of altars, incense, and a priesthood, as in Roman religion; the obsession with blood in Central and South American Catholic artistic traditions being tied to aboriginal human sacrifices in that region of the world; and other similar ideas that often get bandied about by laymen. Do any of these or similar ideas hold water? Why or why not?
I have expressly included Africa and Asia in this discussion so that experts on the Church of the East or Ethiopian Christianity can chime in.
Hey! Sorry for the couple of days delayed response; it's been a bit of a busy week. You've got several questions here, covering over 1500 years of religious experience. I'll touch on the areas I am familiar with, but you may want to re-post some of the sub-questions in the description as questions in their own right.
Luckily, I have already written a fairly detailed post on the origins of the cult of the saints. While it is tempting to interpret the saints simply as the new demi-gods of a new religion, their role and place within the Christian cosmology is crucially different in several important aspects. However, instead of the cult of heroes, many of the early rituals surrounding the veneration of the martyrs and saints paralleled the traditional patronage networks and funerary practices (feasting by grave-sites, vigils, etc.) in the late ancient Mediterranean world.
The preservation of these kinds of elements shouldn't be surprising. Syncretism is a large part of the conversion process, no one completely drops the cultural context that frames them. Frequently you hear arguments that early conversions were surface level and the laity cared little for the new faith, and the persistence of 'pagan' rituals is often pointed to as evidence to that fact. It does not help that church authors frequently bemoan the pagan beliefs and traditions of their parishioners. However, persistent belief in the efficacy of rituals and charms, such as those surrounding birth, are not necessarily indicative of a superficial conversion. 'Christian' and 'Pagan' are not monolithic categories that contain different and mutually exclusive traditions.. Instead, the incorporation of older traditions can simply mean that participants saw no conflict between the old traditions --which do not have to be understood as specifically pagan-- and the new faith, even when such traditions provoked the ire of religious authorities. The reality is, the rituals and traditions that we might call pagan, for folks at time would just been a part of how the world was structured and worked. Pagan after all is a nebulous term that for us covers a broad range of rituals, practices, beliefs, and relationships. Converts would not necessarily recognize these traditions as explicitly belonging to a pagan category of religion and therefore incompatible with the new faith.
Moving on to some of your other questions though:
The Cult of the Virgin Mary has no connection with the Cult of Isis. While Mary is certainly an important individual in Christian tradition from the beginning, the cult did not reach its most developed form until the Late Middle Ages, long after the cult of Isis ceased to be practiced. Were their specific rituals or traditions you had in mind? There are mother-goddess figures in many cultures, but that does not imply a direct borrowing.
I am by no means an expert on Mithraic rituals, but I am skeptical of a direct connection here as well. The two rituals are very different, and ritual feasting is not an exclusively pagan activity; the Eucharistic sacrament is after all tied to the Passover meal very early on in Christian tradition. From what little I know, the Mithraic cult never practiced the ritual consumption of God. Again though, this is not an area I am an expert on. My one caution is to be careful when trying to connect specific rituals. When we make generalizations, it's really easy to discover similarities. But such generalizations are often made at the expense of nuance, where the real answers to the origins and meanings of specific rituals are often found.
Our resident expert on late ancient Christianity, talondearg, covers the relationship between Augustine, Manichaeism, and Platonic Thought more completely here in a more complete fashion than I ever could. To sum up the post, Augustine certainly engaged both in his writings (he rejected Manichaeism outright), but the influence should not be seen as a direct borrowing, but instead as part of a larger discourse of ideas.
I will avoid commenting on Latin American Christianity, as I know little about the matter, but the influence of pre-contact beliefs does seem to be fairly strong. For your other points, the existence of priests, altars, and incense are not explicitly pagan elements, all are featured in Jewish ritual for centuries prior to the advent of Christianity. These things were simply features of Mediterranean religious ritual and practice and not exclusive to one 'set' of beliefs. So yes, I think we can safely say that early Christianity was influence by pre-existing religious traditions, but it is better to frame the situation as a dialogue between relatively fluid religious traditions and not as a 'direct borrowing' or 'appropriation' (in most instances at least).