Constructing a circle through a given point, tangent to a given line, and tangent to a given circle

It is possible, provided that the point and at least part of the circle are on the same side of the line and the circle does not separate the point from the line. If the point and the circle are on opposite sides of the line, or if the point is inside the circle and the line wholly outside, then it is impossible (as it is with three nested circles in the original version).

Wikipedia describes this as Special case 6 of Apollonius' problem with up to 4 solutions

If the circle and the line have a point of intersection, you can send the point of intersection to infinity with an inversion (with respect to a circle which takes this point as center) and you have reduced it to a problem with two tangent lines and a point (which I presume that you know).

For the case without intersection, I would have to reflect further, but in any case, it is equivalent by inversion to one point and two circles if you have already accepted the cases with repetition.