Why bother showing all the ways Patrick lied on the 911 call and in the hours and days after?
I'm not sure he was doing this, I took this to mean that memories created under such traumatic circumstances can be very distorted, and seeing as they used the exact words she apparently said as part of their conclusion that she was criminally responsible he pointed out that the source may not be reliable, but unfortunately did not develop this point properly. It is possible he played into the conspiracy theories a bit there, although the point on memory was a stronger point to the jury and a weak point in the prosecutions case, but unfortunately he didn't develop this point, but if there's a retrial he will.
Why would he say outside court that he wouldn’t believe Patrick if Patrick told him it was snowing in a blizzard?
He was talking about the psychiatrist there, not Patrick.
Why did he show up allllll the ways the investigation was lacking?
Again with this I think it was because the prosecution are saying pre-meditated and pushing murder 1, but to show pre-meditation a more concrete timeline or investigation really should have been done. Like with the crushing of the pills, why didn't they test the cup or try to find what she crushed them with? That surely should have been done, and surely they should have looked to see what she crushed them with, if there was no evidence that she'd just crushed up a load of tablets then wouldn't that suggest that she had crushed them up earlier or another day in preparation? One of the prosecution witnesses made out it was a mammoth task to crush pills and so seemed to be suggesting someone psychotic couldn't do this, but yet there's no evidence whatsoever put forward to prove anything about the crushing or what was used.