A babysitter wouldn't be likely to spread anything to your kid, if that's what you're worried about. Unless the babysitter is also watching other kids. The nanny we had before daycare never gave our kid anything, but did give her something once.
My wife has 3 more weeks before her 12 week maternity leave is up. I'm thinking about buying her a WiFi picture frame and having it set up in her office, loaded with baby pictures.
If you're worried it might make your wife sad, just talk to her about it! Some people like the reminders of kids at work, some don't. My wife keeps a ton of photos on her phone, and cycle through her favorites as phone/lock screen backgrounds.
The only thing I'll warn about the WiFi photo frame is to make sure, however you set it up, it won't ever see or show any pictures you take of your kid in the nude, bathtub, that sort of thing. Some people just set it up to their Google photos account and leave it at that, then get surprised with a funny-to-you picture of your kid being goofy in the bathtub that makes random strangers think you look at CSAM.
Oof. Yeah that's not great. Sinus infections are bad enough, but that final trimester you're kind of on-deck all the time. Like a job you can't take a sick day for. Although in my wife's case, that was also true for her third trimester still working her job.
What books do you guys/your kids like? Mine is recently into Calvin & Hobbes. He can't read yet, so I'll either read them to him or he'll just browse the pictures and start laughing hysterically at the funny faces they make. I kind of think C&H might be a bad influence because he's learning that getting into trouble is fun lol.
Also I don't care for The Giving Tree. My kid likes it ok because of the pictures, but he doesn't get why the story is kind of a downer yet.
My kid (3) likes a lot of different stuff. The current favorites are the Nibbles the Book Eating Monster books, but as a parent doing the reading, it's hard to know what to read and what not (on every page there's usually a lot of text that, if I read it all, she gets bored and wants to move on.
In general, I wouldn't worry about stuff being too emotionally mature or that they'll miss the point. If they love the story, go for it. They will like to come back to it when they're a little older and see a lot more of the depth of the story.