Connor died at Angel's own hand so that he could be "reborn" into his new life, technically fulfilling the once-false prophecy that "the father will kill the son." Other than Angel, and some special cases like Cordelia and Eve, everyone's memories were rewritten to accommodate this new reality.
Towards the end of season four, events take their toll on Connor's sanity, and his memories are rewritten to give him a normal life. The next time he appears in season five, Connor is a well-adjusted person. At the return of his memories, Connor finally reconciles with Angel in the series finale.
Tl;dr: Jasmine possessed Cordelia when she left the higher dimension; the memory restoration only triggered Jasmine's control over Cordelia; Cordelia slept with Connor in the sense that Jasmine was possessing her body at the time; Cordelia's body was pregnant with Connor's baby but it wasn't exactly a baby.
Angel's son Connor is brought back from the Hell dimension a feral teenage warrior, and is hellbent on vengeance against his father for his evil past. After a violent reunion, Angel finds himself battling vampires with his now teenaged son Connor by his side.
Although Gunn and Fred were furious that Wesley hadn't informed them of what had happened and allowed Connor to remain with them, Angel was grateful to Wesley for the rescue. Although offered forgiveness, Wesley refused to return.
In his new life, Connor appeared to be well-adjusted and happy, with an easy going attitude and a sarcastic wit. After his memories were restored and he returned to Angel Investigations, Connor developed into a more heroic figure much like his father.
Wesley is resigned to his fate, believing that he has nothing more to live for now that Fred is gone, and walks away, but not before asking Spike to take care of Illyria. In the new timeline, Angel names a wing of the Los Angeles public library in memory of Wesley and Fred.
Connor blames himself for the whole situation, but Cordelia comforts and reassures him that he's not to blame. Cordelia kisses Connor and offers him the chance to feel something real. While everyone else watches the fire, fearing the future that awaits them, Connor and Cordelia have sex.
In the season finale, Cordelia arranges to meet Angel to confess her feelings, but is prevented from doing so by Skip, who informs her that she has become a higher being. Cordelia accepts her duty, and leaves Earth for another dimension.
However, various events (particularly Cordelia's possession by Jasmine) kept them from admitting their feelings to each other, although shortly before Cordelia died, they revealed their love for one another and shared a single kiss. She died loving Angel, knowing that Angel loved her back.
Wilson and Cordelia kiss again and spend the night together in lovemaking. She wakes the next morning to an empty bed and is horrified to find she has grown very pregnant.
In the end, Angel would bought Cordelia dozens of new clothes (despite telling Wesley he was going to give her time) and was extremely satisfied with her immediate forgiveness afterwards.
Cordelia loves her father as much as any child loves their parent but does not feel she can flatter her father by making him feel there is no room in her heart for any other love.
Connor admits he loves Jenna, to which she says back, and the two kiss.
Angel's ratings in season 5 were actually up on season 4, making the lack of renewal even more of a shock, although the common reason given behind the move is that Whedon pressed John Levin, WB's Head of Entertainment, for an early decision on the show rather than the typical mid-May renewal.
Angel kills the demon. Its blood, eventually found to be the Blood of Eternity, merges with his own, and he becomes human. Realizing what this means for their relationship, Angel spends the night with Buffy.
Cordelia had been taken over by the dark entity Jasmine (Gina Torres), which got pregnant so it could give birth to itself. Season 4 rewrote much of Cordelia's previous character arc, reframing her growth and agency as deliberate maneuvering to bring her under Jasmine's influence.
Ascension into a higher being
Because of this action of accepting demonhood, Cordelia was deemed a higher being and ascended to a higher realm. There she was able to wage the war on evil in a new way as a higher being in paradise with the Powers That Be.
Plot-device demon.
Cordelia does not marry anyone in in King Lear. She remains loyal to her father exclusively throughout the play; if she had a husband, her loyalty would automatically be split.
Canon is that Angel loves Buffy, but due to the curse on him, they cannot be together because if truly happy, he goes back to being Angelus.
Charisma Carpenter, who played Cordelia Chase on Angel, reiterates her character's season 4 pregnancy story arc as problematic in a Tweet.
Angelus terrorizes Willow and the Gang at school, emotionally tormenting Buffy. Later, as they discuss Angel's transformation in the library, Buffy realises that having sex with Angel is what caused him to turn evil.
Angel fires his three colleagues after escalating disagreements about his growing moral corruption. Instead of disbanding while Angel deals with his private issues, Gunn, Cordelia, and Wesley continue running Angel Investigations without him in a rented office.
1. Spike. We know that Team Angel fans are probably screaming at their computers right now, but hear us out. There are plenty of reasons Spike was the most powerful vampire in the Buffyverse.