If you mean can a circle always be drawn round a quadrilateral so that the quadrilateral is enclosed within the circle then yes as long as the diameter of the circle is large enough.
If you mean can a circle always be drawn around a quadrilateral so that it passes through all vertices then only if the opposite angles of the quadrilateral add up to 180o - such quadrilaterals are called cyclic quadrilaterals.
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The ratio is pi/4.
circumscribed means the polygon is drawn around a circle, and inscribed means the polygon is drawn inside the circle. See related links below for polygon circumscribed about a circle and polygon inscribed in a circle.
Always one for sure, and never more than one.
No. Circumvention means to surround or to go around or bypass. It is not a geometric term and has nothing to do with a triangle. Having said that, a circle can be drawn from the circumcentre of any triangle so that it passes through the vertices of the triangle.
The point from which the circle is drawn IS the center.