Can we observe what objects exist at the L3 positiion of planets of our solar system? From what I understand Earth's L3 is obscured by the Sun, so is that true for our viewpoint of the other planets? Or, are there times when other planets are on the other side of the Sun from Earth and we can see their L3 position? For the Hilda's of Jupiter have we observed them in that position or is it that we have calculated their coordinates to be in the L3 position?
You asked


  L3 is obscured by the Sun, so is that true for our viewpoint of the other planets?


No. Each planet orbits with it's own period because each planet is at a different distance from the sun. So most of the time we can see the L3 point of each planet's orbit from Earth. Occasionally it goes behind the sun from our point of view.


  For the Hilda's of Jupiter have we observed them in that position or is it that we have calculated their coordinates to be in the L3 position?


The Hilda family do not occupy the Jupiter L3 point. They generally orbit closer to the sun than Jupiter, and orbit three times for every two orbits of Jupiter. Their orbits are interesting; they have eccentric orbits (up to 0.3) that in some cases cross the orbit of some of the Trojans. You may have noticed that some of the Hildas also approach the Jupiter L3 point. At that point, and when other members of the family approach the Tojans at L4 and l5, the Hildas are moving more slowly. So the Sun's gravity gradually accelerates them, pulling them back to the lower, faster part of their orbits.

I have linked to the Wikipedia Hilda family page. The Lagrangian point page is also helpful.