Hey, I was this kid until about the age of four, and have been cat lady ever since. It may just be something he will grow out of.
When he does any of the above, what are your responses? Does he have any recourse?
For the time being, kitty may want a higher up safe space if their own. While I was pregnant, and was trying to foreshadow events like this I bought two of these
I have four cats so I bought two for good measure. My toddler cannot reach the highest shelves and we can close it off if need be. It's like a little cat bedroom. Mine love to be in them even when my girl is not zooming around pestering them.
after getting to know him the last 3 days, hes very independent and calm for a 4 month old, and very small. hes also very trusting, allowing people to pick him up and pet him with no struggle.
we were considering getting him one of these: https://www.amazon.com/MidWest-Homes-for-Pets-130/dp/B000I1M76Q/ref=sr_1_sc_3?ie=UTF8&qid=1494529393&sr=8-3-spell&keywords=large+cagedcat+condo
so he can have somewhere to stay for the hour or two i try to sleep from 8-10ish. i just, i dont know! he's very enticing but im not sure if im ready for the work.
My device is being funky and won't let me edit, sorry.
Then try 2 feet, then the hallway.
If that is all to much for you, maybe a Kitty condo like this:
https://www.amazon.com/MidWest-Homes-Pets-Cat-Playpen/dp/B000I1M76Q
Cover with a blanket that your DD has slept on for several days. Do the baby food bonding through the bars for a few weeks to win her trust.