It seems to be the only way to remove a pregnancy from a townie (don't ask) as on all the tools I've tried "terminate pregnancy" is only available to household Sims.

Do the abdomen and locomotion animations go away on their own? My ToLD has no such restriction, but I rarely make people pregnant, so I don't know if it works fully.

Yep. All family sims, pretty much regardless of relationship status (but especially in a relationship, and when married), rolls the baby want at some point. It's often one of the first thing they roll. It's also one of the first things they'll roll after having a baby

Big wants like having a baby for family sims are wants that they roll when they are at low (or about to be low) aspiration. Since sims start with absolutely neutral aspiration, that counts as "low" for the system and they start rolling those big wants. If you keep them permanently in platinum, they generally only roll wants for babies when they get married, or after they've had another baby, or one of their children has just aged up.

Is there any special animations for first trimester? I just checked and SimPE does have a Pregnant (Invisible) flag checked but I'm not sure what it does.

I've heard people claim that their sims walk slightly differently in the first trimester, but I've never noticed a difference. I think the belly at least does not go away if you terminate the pregnancy after the first trimester, which is I think why usually the option is only available on modded objects during the first trimester.

If I use the Sim Locomotor to make a Sim walk with a pregnant style, they do that exaggerated walk (which honestly looks hilarious when there's no pregnant belly).

Considering that Andrew mentioned that Cassandra "spectacularly fell out" with Don, that would point to a possible low aspiration.