not a big fan of A/THR but ruling out finger troubles or so-called "wrong levels of automation" as mentioned before i've never experienced any of the first 2 situations described. but yes, A/THR is quite lazy at times and/or doesn't work 100% to plan and sometimes isn't helpful at all in my eyes (i.e. gusty situations). disconnect it then if you don't like it/if out of limits.
A/THR going below VAPP: yes, but i've never seen it going below VLS. since you have VLS+5kt for A/THR ON (or even more for whatever) and considering the PM callouts required for speed deviations (-5kt,+10kt) it's within limits i'd say. but again: if you don't like it, disconnect ATHR and to quote FCTM
If the A/THR performance is unsatisfactory, the pilot should disconnect it and control the thrust manually.