Thanks for this list of possible metrics, lots of things to think about! We don't yet collect enough data for all of these, but some of them we do. A few potentially interesting tables:
Number of searches per session:
+------------------+--------------+---------+
| session_searches | num_sessions | percent |
+------------------+--------------+---------+
| 1 | 59850 | 69.35% |
| 2 | 16564 | 19.19% |
| 3 | 5481 | 6.35% |
| 4 | 2176 | 2.52% |
| 5 | 1001 | 1.16% |
| 6 | 531 | 0.62% |
| 7 | 307 | 0.36% |
| 8 | 204 | 0.24% |
| 9 | 115 | 0.13% |
| 10 | 75 | 0.09% |
+------------------+--------------+---------+
Click through position by number of searches in session:
+----------------+-------+-------+-------+-------+-------+-------+
| event_position | one | two | three | four | five | six |
+----------------+-------+-------+-------+-------+-------+-------+
| 1 | 62.99 | 68.97 | 63.63 | 62.88 | 61.16 | 59.43 |
| 2 | 15.42 | 12.06 | 13.78 | 15.07 | 12.43 | 12.74 |
| 3 | 7.29 | 5.77 | 6.5 | 5.98 | 8.98 | 8.43 |
| 4 | 4.04 | 3.6 | 4.68 | 4.01 | 5.48 | 4.22 |
| 5 | 2.48 | 2.45 | 2.54 | 2.4 | 2.51 | 3.24 |
| 6 | 1.68 | 1.24 | 1.71 | 1.53 | 1.77 | 1.18 |
| 7 | 1.24 | 1.12 | 1.49 | 1.65 | 1.8 | 2.53 |
| 8 | 0.75 | 0.82 | 1.04 | 0.81 | 0.56 | 1.42 |
| 9 | 0.74 | 0.53 | 0.87 | 1.44 | 0.91 | 0.93 |
| 10 | 0.42 | 0.57 | 0.35 | 0.79 | 0.36 | 1.44 |
| 11 | 0.4 | 0.52 | 0.63 | 0.72 | 0.35 | 1.46 |
| 12 | 0.36 | 0.28 | 0.42 | 0.6 | 0.61 | 0.91 |
| 13 | 0.33 | 0.19 | 0.3 | 0.33 | 0.32 | 0.38 |
| 14 | 0.29 | 0.38 | 0.2 | 0.53 | 0.6 | 0.13 |
| 15 | 0.26 | 0.28 | 0.21 | 0.37 | 0.92 | 0.4 |
| 16 | 0.26 | 0.18 | 0.2 | 0.23 | 0.1 | 0 |
| 17 | 0.24 | 0.11 | 0.31 | 0.03 | 0.11 | 0.27 |
| 18 | 0.21 | 0.26 | 0.21 | 0.17 | 0.11 | 0.09 |
| 19 | 0.21 | 0.31 | 0.23 | 0.29 | 0.43 | 0.09 |
| 20 | 0.28 | 0.25 | 0.6 | 0.19 | 0.28 | 0.62 |
| 21 | 0.01 | 0.06 | 0.01 | 0 | 0.03 | 0 |
| 22 | 0.03 | 0.01 | 0.04 | 0 | 0.04 | 0.02 |
| 23 | 0.01 | 0.02 | 0 | 0 | 0.06 | 0 |
| 24 | 0.02 | 0.01 | 0.01 | 0 | 0.04 | 0.07 |
| 25 | 0.02 | 0.01 | 0.03 | 0 | 0.04 | 0 |
+----------------+-------+-------+-------+-------+-------+-------+
Click through position vs duration of search session in seconds. Column's are named as min_max
+----------------+-------+-------+-------+-------+--------+---------+---------+---------+---------+----------+-----------+
| event_position | 0_1 | 2_15 | 16_30 | 31_60 | 61_120 | 121_180 | 181_240 | 241_300 | 301_600 | 601_1200 | 1201_1800 |
+----------------+-------+-------+-------+-------+--------+---------+---------+---------+---------+----------+-----------+
| 1 | 92.52 | 72.07 | 58.59 | 58 | 56.79 | 57.1 | 59.29 | 58.82 | 58.17 | 54.94 | 60.23 |
| 2 | 4.43 | 14.34 | 14.29 | 13.72 | 15.22 | 15.14 | 13.2 | 16.3 | 15.08 | 14.8 | 11.13 |
| 3 | 1.62 | 5.83 | 8.6 | 7.85 | 7.1 | 8.17 | 7.1 | 4.63 | 7.86 | 7.69 | 7.24 |
| 4 | 0.05 | 3.16 | 5.1 | 4.8 | 4.73 | 4.52 | 5.61 | 4.43 | 4.75 | 4.83 | 4.79 |
| 5 | 0.5 | 1.59 | 3.7 | 3.5 | 3.81 | 2.27 | 2.64 | 2.57 | 2.9 | 2.78 | 2.74 |
| 6 | 0.34 | 0.97 | 2.06 | 2.09 | 2.17 | 1.59 | 2.15 | 3.48 | 1.81 | 2.57 | 0.86 |
| 7 | 0 | 0.67 | 1.41 | 1.84 | 1.63 | 2.26 | 1.64 | 1.5 | 2.03 | 2.12 | 2.16 |
| 8 | 0 | 0.27 | 0.9 | 1.52 | 1.32 | 1.17 | 1.39 | 1.1 | 1.28 | 1.46 | 1.28 |
| 9 | 0 | 0.23 | 1.15 | 1.16 | 1.04 | 1.02 | 0.47 | 1.77 | 1.13 | 1.41 | 1.26 |
| 10 | 0 | 0.14 | 0.55 | 0.96 | 0.99 | 0.93 | 0.76 | 0.75 | 0.45 | 1.17 | 0.86 |
| 11 | 0 | 0.17 | 0.41 | 0.76 | 0.66 | 0.64 | 0.53 | 1.23 | 0.73 | 1.31 | 0.99 |
| 12 | 0.03 | 0.08 | 0.58 | 0.59 | 0.74 | 0.65 | 0.55 | 0.19 | 0.45 | 0.92 | 1.32 |
| 13 | 0 | 0.08 | 0.49 | 0.39 | 0.67 | 0.4 | 0.28 | 0.83 | 0.41 | 0.24 | 0.86 |
| 14 | 0 | 0.05 | 0.35 | 0.52 | 0.74 | 0.72 | 0.57 | 0.51 | 0.37 | 0.77 | 0.86 |
| 15 | 0.16 | 0.05 | 0.41 | 0.41 | 0.37 | 0.35 | 0.7 | 0.16 | 0.59 | 0.54 | 0.79 |
| 16 | 0.34 | 0.09 | 0.36 | 0.21 | 0.53 | 0.31 | 0.35 | 0.35 | 0.23 | 0.43 | 0.46 |
| 17 | 0 | 0.07 | 0.17 | 0.32 | 0.18 | 0.56 | 0.52 | 0.31 | 0.31 | 0.24 | 0.46 |
| 18 | 0 | 0.04 | 0.22 | 0.58 | 0.36 | 0.33 | 0.67 | 0.28 | 0.25 | 0.53 | 0.75 |
| 19 | 0 | 0.05 | 0.2 | 0.29 | 0.42 | 0.81 | 0.75 | 0.32 | 0.4 | 0.4 | 0.22 |
| 20 | 0 | 0.02 | 0.44 | 0.48 | 0.48 | 0.44 | 0.74 | 0.36 | 0.64 | 0.53 | 0.31 |
| 21 | 0 | 0 | 0.01 | 0 | 0.03 | 0 | 0 | 0.04 | 0.06 | 0.17 | 0.09 |
| 22 | 0 | 0.02 | 0 | 0 | 0 | 0.12 | 0.01 | 0.01 | 0.05 | 0.03 | 0.24 |
| 23 | 0 | 0 | 0 | 0 | 0 | 0.03 | 0.05 | 0 | 0.01 | 0.07 | 0.04 |
| 24 | 0 | 0 | 0 | 0 | 0 | 0.13 | 0 | 0.03 | 0.03 | 0.03 | 0 |
| 25 | 0 | 0 | 0 | 0.02 | 0.01 | 0.33 | 0 | 0.03 | 0.01 | 0.04 | 0.02 |
+----------------+-------+-------+-------+-------+--------+---------+---------+---------+---------+----------+-----------+
Number of searches performed vs number of clicks on search results per session:
+----------+--------+--------+-------+-------+-------+
| searches | zero | one | two | 3_5 | gt5 |
+----------+--------+--------+-------+-------+-------+
| 0 | 0.00% | 0.08% | 0.00% | 0.00% | 0.00% |
| 1 | 40.85% | 19.77% | 0.27% | 0.05% | 0.01% |
| 2 | 10.56% | 7.86% | 1.46% | 0.06% | 0.00% |
| 3 | 4.75% | 2.55% | 0.98% | 0.27% | 0.00% |
| 4 | 1.91% | 1.25% | 0.51% | 0.29% | 0.00% |
| 5 | 1.14% | 0.55% | 0.30% | 0.23% | 0.01% |
| 6 | 0.61% | 0.31% | 0.17% | 0.18% | 0.02% |
| 7 | 0.37% | 0.19% | 0.11% | 0.16% | 0.02% |
| 8 | 0.24% | 0.10% | 0.08% | 0.10% | 0.02% |
| 9 | 0.14% | 0.06% | 0.06% | 0.07% | 0.02% |
| 10 | 0.11% | 0.06% | 0.03% | 0.04% | 0.02% |
| 11 | 0.07% | 0.03% | 0.01% | 0.04% | 0.03% |
| 12 | 0.06% | 0.02% | 0.01% | 0.02% | 0.02% |
| 13 | 0.03% | 0.01% | 0.01% | 0.02% | 0.01% |
| 14 | 0.03% | 0.01% | 0.02% | 0.02% | 0.01% |
+----------+--------+--------+-------+-------+-------+
This isn't nearly all the things you mentioned, but a few that we have collected data for already. Hopefully these and a few other metrics can become part of how we measure what effect changes to search make.