Is it possible for the ignition coil to melt to the wall of a spark plug

050320

About a year ago I got a new battery and had a tune up done. He was unable to change 1 spark plug. He says the ignition coil melted to the wall of the spark plug.

Since then the car has run okay. Yesterday it didn't start. No sound or anything when I turned the key. Some one gave me a jump. Until that moment there was no indication that the battery was getting weak.

I was then able to drive a few miles, park, turn it off, start it and drive another 1/2 mile and park.

I removed the battery so I could take it into the store for testing. They said it was good. I reinstalled it. I started the car. But then it sputters and shuts off immediately.

I found that by starting in neutral and sliding into gear fast, I could get moving. I drove about 1-1.5 miles. The car idled a bit rough at low speed and stops but didn't shut off until the last half of my trip. Then it shut off a few times at stops.

So I have 2 questions:
1. Can an ignition coil melt to the wall of a spark plug, and if so, would the car continue to start and run for a year?

2. Why would my car start sputtering and shutting off as if its not getting fuel when at rest after I removed and reinstalled the battery? It wasn't happening before I did that.

Side notes that may or may not have a bearing on these questions:

1. About 12 days ago my car overheated. I had to replace the radiator.

2. About 1 week ago someone hit my front end. He made a left turn into my car. Damage is bad, but I don't think the engine was damaged. Of course, I don't know. Mostly it looks like the bumper, headlight and hood damaged.(See pics)

Thank you for your help.