Don't have these problems on an Elan touchpad in an Asus laptop.
1. I don't recall clicks that don't register, though I very rarely use physical clicking. No separate buttons here, but I anyway much prefer a gentle tap to actual clicking.
2. Maybe related to settings that temporarily ignore the touchpad while typing? Figure out whether your touchpads are Elan or Synaptics, try to find/get to work alternate/their generic drivers and see if they expose more configuration options.
Elan's been a pain in this regard, with no official 1st-party drivers available, and OEM customized drivers that are always nerfed or tweaked in different ways, including registry settings that don't show in the GUI and unknown ones that change the overall pointer movement feel. In many cases they're also made difficult to install on PCs not from the specific OEM.
(It's one big annoying mess. After having to reinstall once, I gave up on tweaking everything to perfection, and also lost a few features I used to use. I still have the 14 driver variants I downloaded back then, but I didn't take down full notes so I don't know which one was the best, nor what were all the registry tweaks.)