What should I do about partially tacky urethane finish?
It's possible it's just the cool temperature and it will dry in a few days.  It won't hurt to let it sit several more days and see if the tacky areas harden up on their own.  

However my suspicion would be a compatibility problem between the urethane finish and the "green" mineral spirits.  Although they probably say then can be substituted for real mineral spirits, I've heard of mixed results when using them for thinning oil based finishes.  For a future project you could select all water-based finishing products which would have lower VOCs and would not require thinning with any solvents.  Or use standard mineral spirits for thinning and save the "green" ones for cleanup.

If the finish doesn't harden on its own I would try to remove most of the finish from the tacky areas as you described.  I would do it in stages with some drying time inbetween because if you remove too much you will likely also remove some of the stain resulting in inconsistent shading.  In that case you may need to strip and refinish a larger area (or live with the spots).