Road traffic would have generally been less substantial and speeds would have been lower, but accidents could definitely occur. There is one form of carriage - the phaeton - that actually had a strong reputation for being quite dangerous and difficult to handle. It is frequently depicted in the literature of the era as a rather evocative yet fickle mode of transportation. Tipping over a phaeton going round a corner would be a bit like oversteering a Porsche 911 into a ditch.
Absolutely. They certainly had less potential for damage, but they were far from unheard of, especially in busier streets and in tight confines/on certain terrains. As /u/eighthgear notes, the technology itself was also a significant source of injury. Mechanical failure wasn't uncommon, and the fact that the coach was attached to a horse for locomotion made the implications of this far more severe than it otherwise could have been. Here's one example that sums this up pretty well, from the life of Mary Lincoln:
The suffering Robert alluded to came from Mary’s July 2, 1863 carriage accident, during which, while riding into Washington from the Soldier’s Home, the driver’s seat of the presidential carriage became detached, throwing the driver to the ground. The frightened horses began a frantic run along the Rock Creek Road, near the Mount Pleasant Hospital, and Mary leaped from the carriage to save herself. The news reports of the incident stated that Mary was stunned, bruised and battered, but no bones were broken and her injuries, which were immediately administered by surgeons from the nearby hospital, did not appear serious. She did suffer a bleeding wound on the back of her head caused by a sharp stone, which doctors at the nearby hospital quickly stitched up. The president telegraphed Robert at Cambridge not to worry, “Your mother very slightly hurt by her fall.
http://clevelandcivilwarroundtable.com/articles/mary_madness.htm
Speed, disrepair of roads, overcrowding, weather, and top-heavy coaches also posed significant threats to safety in England in the early modern period:
But in bad seasons the traveller had to encounter inconveniences still more serious. Thoresby, who was in the habit of travelling between Leeds and the capital, has recorded, in his Diary, such a series of perils and disasters as might suffice for a journey to the Frozen Ocean or to the Desert of Sahara. On one occasion he learned that the floods were out between Ware and London, that passengers had to swim for their lives, and that a higgler had perished in the attempt to cross. In consequence of these tidings he turned out of the high road, and was conducted across some meadows, where it was necessary for him to ride to the saddle skirts in water.(136) In the course of another journey he narrowly escaped being swept away by an inundation of the Trent.
Among such contributors to poor safety were the emergent flying coaches as a common mode of transportation around the Restoration period, albeit not without protest from the upper sorts. Take the following:
At the close of the reign of Charles the Second flying carriages ran thrice a week from London to the chief towns. But no stage coach, indeed no stage waggon, appears to have proceeded further north than York, or further west than Exeter. The ordinary day's journey of a flying coach was about fifty miles in the summer; but in winter, when the ways were bad and the nights long, little more than thirty. The Chester coach, the York coach, and the Exeter coach generally reached London in four days during the fine season, but at Christmas not till the sixth day. The passengers, six in number, were all seated in the carriage. For accidents were so frequent that it would have been most perilous to mount the roof. The ordinary fare was about twopence halfpenny a mile in summer, and somewhat more in winter.
Source: T. B. Macaulay's History of England, Vol. I Ch. 3