That does not look promising to me but @DannyP has experience making a Jamar rod shifter work well, so he may have better advice.
If that were my car I would remove that stuff and replace with a Brandwood Cable Shift. I believe it can be retrofitted with the engine and trans in the car, and it's comparatively easy to adjust once installed.
Caution though: the untold drawback of the Brandwood is the shifter enclosure, which is a bit larger than the others. In a Spyder that matters both for aesthetic and practical reasons. The good news is you can cut down (or fold) the box to reduce its height considerably. That's what I did and I recommend the procedure to anyone installing that shifter in that chassis. (Your stick looks suspiciously like the one that came in my Brandwood, btw).
Your floor is heavy duty fiberglass. You can bolt your seats (and shifter enclosure) to it and it's fine. Use big fender washers. It's not a jack point.
Jack points: Look where your frame is. The floor is riveted to it underneath. Put a hard rubber or wood block between the jack cup and the rivet line to jack it up. You can also lift it by the beam in front and any of the steel crossmembers in back. Every part of the frame is plenty strong.