Intermittent rough idle when starting and warm

My car was sitting unused in a garage for about two years. I had to replace the battery, and for some reason also the alternator was bad after that time (had no issues when it was parked).

I drove it to my friend's repair shop to get the alternator replaced, which he did. He advised me to get new gas in it and apply injector cleaner, which I did. I also applied Heet. On the way to the gas station, there was thick black exhaust, which I attribute to bad gas, because I never saw it again. I drove it back to his shop where we discussed the exhaust; he had me press on the accelerator while he was looking at something under the hood.

That is when he noticed that the heat gauge was off the charts. He had me shut it down immediately. It turned out that no coolant had been in the car the whole time, apparently it had evaporated while sitting. He had never checked it, and I would not have known to do so.

Coolant was poured in, resulting in jets of steam. This went on for a while until it was cool enough to be filled with coolant. When it was restarted, there was a terrible problem with the idle. He detected misfires on two cylinders and thought that there was serious engine damage.

The car seemed to be a loss. But later that day, he started it again, and it was completely fine. He drove it to my house. However, there is not a happy ending yet. I have a terrible issue after I have driven it a while, switch it off, and again restart it within a short time. I get issues with a very rough idle which will either 1) 'catch' and then the car will be fine to drive, or 2) idle rougher and rougher, until it bucks (the entire car will jolt forward and back), and eventually stall. Once I saw white smoke coming from the hood. Again, I thought that there was permanent engine damage, but upon starting the car again a couple of days later, again it was fine.

So this is a very strange problem, which affects the start mostly when the car has been recently used and driven for a period of time (a warm start). It will idle rough, 'catch' and then idle and drive fine, or it will degenerate into a really bad state, as I have described.

Please help. My mechanic friend is at a loss. I do not know enough about car mechanics to hazard a guess.

Thanks.