Engine starting

Sounds a lot like me. Your description sounds like it needs a throttle body cleaning. Dirt/carbon build up in the throttle body bore and block the air that allows the engine to idle. It also makes the engine run rough when cold because the computer thinks it has a certain amount of air at a certain throttle position. With the build up it no longer does. So the cold start settings for the throttle no longer apply.

Cleaning it involves removing the throttle body and using cleaner and brushes and rags to clean the deposits.

This is a different vehicle but it shows the build up and the cleaning process.
https://www.2carpros.com/articles/throttle-actuator-service

Make sure you get throttle body cleaner. Some of the generic carburetor cleaners can damage the coatings that are used inside the bores on some makes.

Good to hear, keep us in the loop if you don't mind.

It wouldn't hurt to see if it has some codes hidden. It may also show that you have a failing sensor like the MAF or low fuel pressure from an intermittent issue. That's the hard thing when it isn't a constant problem.

Steve W. Is one out best! Use 2CarPros anytime, we are here to help. Please tell a friend.

Cheers, Ken

Check the fuel pressure regulator for leakage, There are two versions depending on when it was actually built. One will be on the fuel rail on the engine. The other is mounted under the veh. On the frame. The pictures show them. They operate the same way and can fail the same.
Both have a vacuum hose attached to them. Remove the hose before you try to start it cold. If you see ANY gas in the hose or evidence of gas the regulator is bad. A small leak could cause both of your problems as it would let a small amount of fuel bypass the system and then clear up as you opened the throttle.

Check all of the vacuum lines as well. A small crack or damage could cause an air leak that would improve as the engine warms up.

I would also run an injector leak-down test as a small leak in on of those could also cause a rough idle but get better with a bit of throttle.

Good to hear you found the problem.

The hose would cause a large vacuum leak. The reason it causes more issues when cold is that the extra air causes a lean mixture that doesn't like to ignite, that same mix will light better when the engine warms up. Same issue was common on carbureted cars if the choke didn't work properly, the lean mix wouldn't ignite.