Logan was a terrible movie and as unnecessary as Terminator 3.
The fate of Logan and Professor X were perfectly wrapped up in X-Men: DOTFP.
Logan had peace, a family, and Jean Grey alive, while Prof X saw his lifelong dream fulfilled. It brought a perfect closure to the story arc of the prior films and had all the requisite cameos and fan references.
I think
@Redux91 answered this part very well.
Not only does Logan mindlessly restore the bleak future the X-Men fought so hard across several movies to prevent, it doesn’t even makes sense in the context of events. Those young kids who were running around in DOTFP apparently couldn’t have even existed since no mutant births happened post 2004. Logan, who looked fine a few years ago suddenly succumbs to adamantium poisoning and every other mutant on earth vanishes.
Logan I used to believe takes place in a parallel timeline from the X-Men "prime" universe. My understanding that there were no more mutants born after 2004, which is around the time the dam broke at the end and Jean Grey died. They mentioned this in Logan where whatever compound they were developing to suppress the X-gene had gotten into the water.
However, I will admit the timeline gets muddy as the samurai sword from The Wolverine was present in Logan. This would mean both DoFP and The Wolverine are part of this timeline, which is the "corrected one". Meaning, X-Men 3 was never made.
So basically, the chronology is X-Men 1, X-Men 2, X-Men 3 (then removed), The Wolverine, DoFP then Logan.
Anyhow, as for the kids running around, they were the result of genetic manipulation. I don't think they were "born" mutants, but I could be wrong. It is also important to mention that I think the X-gene was suppressed, not eliminated.
Logan takes place in 2029. Assuming if DoFP is canon in this universe, that ends in 2023. It is possible Logan himself may had been getting sick from his adamantium then, but even if he appeared fine, that would be 6 years prior. I hate to bring in real-life topics, but my uncle passed from ALS. He appeared completely healthy in 2010, but just plummeted in 2011 to 2013 until he passed. Switching back to fiction, I think it is plausible Logan took time to get sick but I guess his healing factor took a beating. Also, if DoFP is canon, that means The Wolverine was also and his healing factor took a beating there as well.
Let’s face it. They s*** on the prior film in order to forcefully insert an unneeded, hackneyed, EMOTIONAL(TM) death scene. It was a long, exhausting, nihilistic film with a hackneyed ending.
I actually though the final death scene was perfect. If you watch the last lines of Yukio to Logan, she predicts exactly how he is going to die. I'm paraphrasing, but it was something along the lines of being covered in blood with his heart in his hands. That was a metaphor as his heart was his daughter.
Anyhow, I initially was mixed about Logan when I first saw it since when it comes to X-Men, I am used to the spectacle. However, I eventually appreciated it and thinking about the nods to his character's history and overall arc, I thought it was well done.