Hesitation and pulsing rough idle on cold start?

2012 Honda Civic, 165k miles. Been having a lot of hesitation off the line (picking up speed from a stop). When slowing down from 5mph to 0mph, and while stopped, my idle is rough and "earthquakey" in bursts.

On cold starts my car idles rough and feels like a pulsating feeling like the car's going to die. RPMs on cold start are 1500 and take 10minutes to drop to 800.

No codes, shifting is smooth. Any ideas?

Won't start not sure if water pump needs replacing or needs new engine? 2004 Honda civic DX 1.7L engine 155k?

My car won't start i had to press the gas really hard to get it going before but now it started smoking(white) from under the hood and smelling burnt every time we try and start it up. I saw leaking under the car. Probably the coolant because its blue. Not sure if a new water pump and timing belt would do the job and or if we need to swap the entire engine. Before this I noticed that the engine would get hot or shut off whenever we stopped moving and finally this happened.

Can you have 2 cars in one name?

I'm car hunting, and I'm debating on getting 2. A honda civic to use as a daily driver and a 91 camaro to drive for fun. My question is, can I register them both under my name with me as the primary driver?

Replacing PCV valve in a 2012 Civic?

When replacing the PCV valve in a 2012 Honda Civic, is there any potential for oil to leak out while replacing it?

Not sure if I can just unscrew the PCV and replace it, or if I need to do some sort of prep beforehand to prevent oil from escaping?

Pic here: Replacing PCV valve in a 2012 Civic (PCV valve is "1").

Could antifreeze be in the engine?

Recently I had an issue where my 1996 Honda Civic temp gauge started to rise. I stopped the car to let it cool. I got to my destination and called husband to bring antifreeze to fill it back up. I drove it home that night and parked it. Found the problem was the radiator cap. Well now I'm noticing when I start the engine it sounds like it struggles and the RPMs rise higher than normal. It idles fine after a moment. Could this mean antifreeze leaked into the engine?