Is the carotid angiography dangerous to do, as it's near the brain?

Small risk. The risk of stroke is less than 0.5% so it is small but there is risk. Other risks like other angiograms related to access site complication like bleeding or contrast reaction related.
There are risks. Angiography is an invasive procedure which may be used anywhere in the body. There is always a small risk of complications (usually less than 5%). Carotid angiography when performed by skilled hands and in a good center has a low risk. The risks are outweighed by the potential benefits to the patient with cerebrovascular disease.