The camera on the Note 4 is one of the best available.
DxO Labs, a French company that anyone who's a high end Amateur or Pro would most likely name as THE authority on Cameras & Lenses, ranks it as #2 in the World currently. (The S6 taking 1st). They place it slightly ahead of the iPhone 6 Plus, a KILLER camera.
OOPS - TRIED FOR A LINK, BUT I'M NOT HERE ENOUGH ... Just search "DxOMark Note 4" & look for "Hitting the Right Note"
However, the tremendous demand by us for top notch cameras in our phones has lead the companies to start shoving the technology usually reserved for Prosumer or Pro level DLSRs into the hands of the average person. They TRY to make it simple, but it just isn't. I'm NOT knocking the OP's skill level, as I've only had my Note 4 for a few days and haven't LIVED with the camera yet, but people tend to think that having a high end camera makes them a pro photographer.
Having a $2000 knife collection in the kitchen doesn't a Chef make.
EVERY Camera, from the smallest cheapest point & shoot to the $40,000 monsters all have their learning curve and requires time to discover and account for their idiosyncrasies. They ALL have their drawbacks, even the $40K ones!
I SEE the flaws in the posted pics, and I GET what you're saying, but this is a camera in a device that is a COMPUTER AND PHONE 1st, with a camera function thrown in. There are many reasons for lag in such a device, most of which has ZERO to do with the camera itself.
BTW: The 1st Dog pic's main problem is the DOG MOVED. Shutter speed control will help, but you have to know WHEN to use the feature, and the impacts it will have on other aspects of the photo.
Experience with the camera will help almost all the problems you had, knowing what mode to use, and when. Ask
any Pro what percentage of their shots are just C**P, and they'll laugh and tell you "Way to many, Dude, WAY too many". The Pro know you just gotta pound at it.
I'll second the thought that for the dog pics, Sport mode would help. All those modes are there for a reason. We're not at the "Well, the Camera will figure the scene out and adjust for it" point yet. It's a heck of a lot better than it was, but it still requires you to choose the proper mode ...

None of us
ever stop learning!
Rumor line has that the 5.1 upgrade is primarily to add stability and improve performance in many areas, I'm sure the Camera Module will be improved as well.
It's also thought that they will add an Exposure slider like the iPhone has. One more feature begged for by the photogs, one more rock in the path of the untrained.... ;-) LOL
Peace out!
PuterPro