What if people could have children without women giving birth? I'll mostly ignore for the moment related things like living forever or choosing exactly which combination of genes forms the baby. This is just growing the baby outside the body.
Somehow you get a fertilized egg from a mix of the parents' cells. The child under construction is basically a sack of water that needs oxygen and nutrients and heavy padding. There are many form factors for that. An bird egg, for example. The machinery could become compact and widespread, so it could still be a DIY project for parents, no need for dependence on the state or corporations. After term, breastfeeding is still an option, or not, same as today. Raising young children and teenagers is still a hands-on one-on-one job ideally done by multiple dedicated adults.
I asked some females, if you could have children without giving birth, would you still want to give birth? They unanimously answered yes, they'd still want to carry the child in their tummy and give birth. They were keen on birth not being quite so painful and life threatening, but otherwise they wanted to keep pregnancy. The baby listens to your heartbeat. There's a bonding that wouldn't happen otherwise. (They also said I must be reading weird science fiction in order to even think of asking such a question.) Actually, we're already sort of in this realm, if you count cesarians as not giving birth.
Let's follow this egg idea anyhow. Early pregnancy, you have what looks like a 10-pound ostrich egg that you can keep plugged into the incubator at home, or you could carry around. Later term you'd have to upgrade to a 20-pound, 30-pound, maybe 40-pound egg, depends how compact the biological machinery is that keeps the baby warm and healthy and growing. I imagine a major consideration is artificial lungs, to keep it supplied with oxygen. Maybe the egg can extract it from the atmosphere, or maybe there's a time limit for how long you can keep the egg unplugged.
The shell is just a wrapper. You could paint it like an easter egg. You could instead have a leather shell, or suede, or metal, or fuzzy, or with tire treads. You could have it shaped like a box or a bean or a duffle bag. You could embed it in a robot that can walk around like a human or a dog ... but preterm babies aren't thinking yet, heck six-month-olds hardly are, so you can't have the baby decide itself where to go. If the parents don't keep it in the incubator, they've got to keep track of it themselves.
You could carry the egg in a sling so it hears your heartbeat and you can talk to it and it can feel the motion of you walking around. You could hold it in your arms, or push it in a stroller through the park. You could dress it up in fuzzy hats. You could prop it in front of the TV and play Sesame Street and Mozart at it.
If having eggs became the norm, after a few generations women's ability to give birth and even become pregnant would go away. (Even today many women can't give birth. But this way those women would have children anyhow, so soon most children would have parents who can't give birth, so they'd inherit the inability to give birth.)
The shape and growth rate of preterm humans is strongly determined by what's needed for a successful birth. With those constraints gone, preterm humans would probably stay preterm longer, and grow bigger and with bigger brains before term. My impression is that babies and parents would be better off if babies were born at about the current 3-month-old mark (er, one year after conception rather than nine months). So, babies would also quickly lose the ability to survive birth.
This will split the human race into two species: those that can give birth and those who can't. Both could have an egg, but only male+female of those who can give birth could give birth. A male who can't give birth mating with a female who can give birth may give pregnancy, but birth would have to be by cesarian at best, and quite likely life-threatening. Cesarians alone could split us into two species.
The near-term future of sex. Women wouldn't become significantly more interested in sex just because they couldn't become pregnant. We have effective birth control already so women's motives would be about the same as today. I don't know how many generations men's over-the-top sex drive would last if having sex did not cause children. Rape may decrease (less motive) or increase (not punished as severely because it can't force a pregnancy). Men's sex drive is a large factor of getting them to talk to women at all and keeping parents together long enough to raise a child, and raising children is still required. So men's sex drive will remain near-term too. Men would develop more of women's biological clock, since committing to raising children would reliably cause children.
The whole reproductive system ... ovaries, testicles, estrogen, testosterone, uterus ... is stressful on humans. Eunuchs tend to live long healthy lives. Sex organs are a common cause of cancer. It seems likely that these would decrease if they weren't strictly needed. On the other hand, traits useful for attracting a mate would continue. I imagine homosexuality would increase, if a parter of the same sex is just as effective at producing children as a partner of the other sex.
Long term, without the mooring of a functional requirement, the sexual act would drift in unexpected directions or disappear entirely.