1. Regarding a/c climb (and descent) I have this high rate often. It seems in order to regain VNAV path the a/c will dive or climb at 6000 fpm to achieve! This regularly then overshoots the desired path and cannot be halted, only by disengaging the auto in most cases (as you found). FLCH doesn't help as an alternative, V/S can. I will test more and enter topic later.
2. Steering is very sluggish and if off course on close final very hard to recover. I have checked my Yoke settings, all good.

3. Sometimes you will only see the G/S in the PFD less then 5 miles out. The VNAV path indicator must be used. Again I will test more.

Keith