Use the Delineate Tin Data Area tool (in 3D Analyst tools / Data Management / Tin) to get rid of those nasty long triangles in the no data areas of the tin. Choose a distance more than your data spacing and use the options "PERIMITER ONLY".
To clip a TIN based on a polygon, use the "Edit TIN" tool, which is available with "3D Analyst" and the according license.
Set your polygon as the input feature class and select "HardClip" for the "SF Type". The result is a TIN whose extent is exactly the extent of your polygon.