Well, this a really big question, and I'm not the most qualified person to answer, but I can definitely give you at least somewhat of an answer.

The two main still existing branches of Buddhism are Mahayana and Theravada.

Mahayana spread around China, Japan, Thailand, Burma/Myanmar, Laos, Tibet, Vietnam, Nepal, lots of others. Mahayana Buddhism encouraged monastic lifestyle and ideals, and is early on mostly used as the description of these rules or doctrines for bodhisattvas on the path to enlightenment. As a side note, we know that there was at least some of this distinction between branches of Buddhism from a Tang Dynasty Buddhist monk named YiJing who visited India in the 7th century and made some of the first documentation of the differentiations. Essentially, enlightenment is a path to be walked, and can be achieved through working along the path. This is what you see in Tibet and where even more divisions come along (that I'll talk about in a second).

Theravada is the older branch, and both Mahayana and Theravada have a lot in common, but the main differences are: Theravada does not believe that there can be backward progress towards enlightenment, and enlightenment is a sudden phenomena rather than one attained through monastic life.

On to China for a bit. As far as how the connections between Daoism, Confucianism, and Buddhism (Mahayana really) were viewed, well... let's talk about China. Yijing, as mentioned before, was a pretty prominent Chinese Buddhist monk. As Buddhism was brought into China, it didn't remain terribly pure to the original iterations though, as lots of local indigenous practices were brought in: essentially, there wasn't an abandonment of local practices, but a merging between the new religion and traditional practices (more focus came onto ancestral worship and worship of beneficial deities as opposed to the original Indian/Hindu class emphasis). As further examples of the changes between "Indian Buddhism" and Chinese Buddhism, early translators intentionally referred to Buddhist ideas by Confucian or Daoist terms. Dharma was referred to as the Daoist concept of dao (道, the way), the state of nirvana (ultimate bliss) as Daoist wuwei (無為, the concept of doing without trying, kind of), sila (ethics/morality basically) as Confucian xiaoxun (孝洵, filial piety), etc. Still, it wasn't a majority of Chinese that converted, as the population still mainly followed Confucianism and Daoism. Actually, some Chinese actually mistook the Buddhist form as a form of Daoism, to give you an idea of the amount of changes from the original form. So, a traveling Mahayana or Theravada practicer may not have recognized the forms in China.

Some Tang officials took to Buddhism as an intellectual pursuit (some aristocrats, on the other hand, provided land for Buddhist temples purely as a sort of tax shelter, leading to problems later on). Others, not so much. Han Yu was a confucian scholar in the Tang government who was very critical of Buddhist growth in China (he was critical of Daoism too, but he at least acknowledged that Daoism was Chinese in origin). In 819, the finger bone ("supposedly") of the Buddha was circulating around China with the Emperor's blessing. Lest the masses get confused by this strange alien practice, Han Yu implored that the Emperor put an end to all this (directly to the emperor):

Now the Buddha was of barbarian origin. His language differed from Chinese speech; his clothes were of a different cut; his mouth did not pronounce the prescribed words of the Former Kings, his body was not clad in the garments prescribed by the Former Kings. He did not recognize the relationship between the prince and subject, nor the sentiments of father and son.

So, to sum it up, Confucian-Buddhist relations were pretty strained at best, and Buddhism in China was officially repressed and diminished at several points.

Skipping back to Tibet, there are a few evolutions and changes in Buddhist thinking, so it's not even viewed as a single entity there either. Generally, the time is split into three parts: The First Propagation, The In-Between Period, and the Second Propagation.

In the First Propagation, we have the introduction of Buddhism by Songtsan Gampo and the building of the famous Jokhang temple around 650 AD. Then, with Padmasambhava in the 8th century comes the introduction of a tantric form of Buddhism, in contrast to Mahayana and Theravada, called Vajrayana. In Vajrayana, there is the idea that transmission of teachings is a personal experience, between only teacher and student, not through self study. There's quite a bit more, but also monks were non-celibate and there is a large focus on physical actions and activities as expressions of enlightenment.

In the In-Between period also spawned a tantric practice: the practice of Chod. Chod was introduced around the 11th century, and emphasized the body as a tantric feast (weird already, right?). It's a bit different than the monastic life of Mahayana, in that it encourages a cultivation of fear and pain to cut through attachment to one's self, and grow the emptiness inside on the path to enlightenment. Chod practitioners often hung around graveyards and caves and such, as isolated practitioners reveling in the darker practices and had a whole slew of separate symbols (for example, often you'll see Chod practitioners with a knife as a symbol of cutting through the ego). During the In-Between period, there was much argument about whether to follow Mahayana practices, Theravada practices, and the status of tantric practices (again, there's debate on what's really Buddhism). It was a major formative period where Buddhism in Tibet came more into it's own.

Finally, there's Second Propagation during the 10th and 11th century. During this period, many tantric practices were dropped in favor of a return to more of an Indian style Buddhism. Yeshe Od of the Tibetan Empire, abdicated the throne, and in doing so denounced sexual yoga, animal sacrifice, and advocated a return to the older orthodoxies. Following, Attisa promoted monastic discipline and the reemphasis of the gradual path offered by Mahayana as opposed to the instant enlightenment sponsored in Theravada or the hastening of progress in tantric practices.

So, essentially: Buddhism is definitely a broad term and does not accurately reflect all the subdivisions, you are correct. Daoist relations are pretty minimal, except that some traditional Daoist practices and ideas found their way into Chinese Buddhism. Most Confucians were not fans of Buddhism, but the association in translations was a matter more of convenience than true similarity or shaping. Buddhism (in both Mahayana and Theravada) became pretty regional as opposed to being universally constant.