I'll address the last question first. A lot of non-Christian (and, in fact, uninfluenced by Christianity) traditions do purifications before they do ritual. Siberian shamans (Ulchi, Nanai, etc) purify with smoke. Lakota people purify with smoke. Shinto purifies with water. Many other traditions to purifications of varying kinds. They do them for a lot of different reasons.
Purifications like this don't imply that we are evil or sin-filled or unfit to stand before the Gods. A pre-ritual purification can be viewed in the same way as wiping your feet before you walk into someone's house or taking off your shoes at the door -- sometimes we carry stuff with us that we don't want to bring into a ritual space on our bodies or our clothing. It's not a value judgment about us as human beings, it's just a way to remove any unwanted influences before we enter ritual space, however we are defining that.
The reason I use juniper in particular is because it was used in Scotland. We don't know if they did this every time anyone did ritual, but for a lot of people, doing something like this before ritual is a comforting thing, or a signal that we are entering ritual space. In Scotland, juniper was burned on New Years morning in enough quantity to fill the entire house or byre. I don't want to suffocate my whole house and make all the smoke alarms go off, so using a little of it at the beginning of a ritual was something I considered a reasonable adaptation of the tradition. It was also burned on the quarter days (Imbolc, Beltaine, Lughnassadh, Samhain) for the same reasons.
Here is a quote from F. Marian McNeill's The Silver Bough (link to ebook edition) regarding that tradition:
Juniper, or the mountain yew, was burned by the Highlanders both in the house and in the byre as a purification rite on New Year's morning. Like all magical plants, it had to be pulled in a particular manner. The Druids, as we have seen, had considerable medical skill. They knew all that was known of botany and chemistry, and to them fell the selection of the herbs for the mystic cauldron. These were gathered at certain phases of the moon. Magical rites were employed in the culling; sexual abstinence, silence, a certain method of uprooting, and occasionally sacrifice were necessary. Long after the disappearance of the Druids, herbs found by sacred streams were used to cure wounds and bruises and other ills, and traces of the rites and runes linger in folk tradition. Juniper, for instance, to be effective, had to be pulled by the roots, with its branches made into four bundles and taken between the five fingers, whilst the incantation was repeated:
I will pull the bounteous yew,
Through the five bent ribs of Christ,
In the name of the Father, the Son, and the Holy Ghost,
Against drowning, danger, and confusion.
As with so many Gaelic prayers, we can take the form and consider which of the Pagan deities would suit the situation. Miach seems a very reasonable deity for the second line here, given that all of the healing herbs were said to grow from his cairn. The third line might work well with Dían Cécht, Miach, and Airmed, if you are so inclined.
John Gregorson Campbell, in several places in The Gaelic Otherworld, says:
Juniper, pulled in a particular manner, was burned before cattle and put in cows' tails.
Juniper (Iubhar-Beinne, literally Mountain Yew): This plant is a protection by land and sea, and no house in which it is will take fire.
Shrovetide [the Tuesday before Lent] was one of the great days for 'saining' cattle, juniper being burned before them, while other superstitious precautions were taken to keep them free from harm.
The Carmina Gadelica offers:
Iubhar beinne [juniper] and caorran, mountain ash or rowan, were burnt on the doorstep of the byre on the first day of the quarter, on Beltaine Day and Hallowmas. The byre lintel was sprinkled with wine, or failing wine, with human urine. ... This was done to safeguard the cattle from mischance, mishap, and each other's horns.
Milliken and Bridgewater, in Flora Celtica: Plants and People in Scotland say:
Juniper is another tree whose branches were sometimes hung above the doors and windows on auspicious days, or burned in the fire. Juniper burning, which formed part of the New Year rituals in some parts of the country, seemed to have a dual purpose. Not only was it supposed to ward off witches and evil spirits but, at a more practical level, it cleansed the house of pests and diseases. The branches were dried beside the fire the night before, and when all the windows and doors were shut, fires were lit in each room until the whole house was full of their acrid smoke. When the coughing and sputtering inhabitants could stand it no longer, the windows were opened and the process was repeated in the stables. Interestingly, the smoke of burning juniper is also used for spiritual cleansing in Nepal, where it plays a key part in puja ceremonies such as those held before attempts to climb Mount Everest.
I don't have any particular investment in warding off witches, given that a lot of my friends fall into that category, but warding off evil spirits, bad luck, illness, danger, fires, and general klutziness seems like a pretty good reason to follow this tradition. Besides, juniper smells wonderful, and it grows abundantly around here, just as it does in the Highlands of Scotland.
Juniper was also known in Irish and Gaelic as aiteal. This Irish gardening website claims:
The stems and branches which provide support for the trees foliage and berries are covered in rich, brownish red bark, which can be seen to shred, curl and peel away in strips from the mature tree. Under the bark, you will find the pinkish white water-filled sapwood similarly aromatic to the pungent foliage. The interior brown heartwood is quite soft and has few if any wood working uses, apart from veneering; instead, it was used for burning because of its scent. The ancient Celts burned the wood of the Juniper at their autumn (Samhain) festival for purification, as an aid to allow contact with the dead.
I haven't seen other references to the Irish or Scottish use of juniper for contact with the spirits of the dead, but this may well come from traditions in other parts of the world and have been attached to "ancient Celtic" uses for the plant, given that it was burned at the quarter days, one of which is Samhain. If anyone has further information about where this particular reference might have come from, please let me know. I'd be interested to see the sources on it!