It will not restart unless I wait at least 15 minutes, then it will start and then it runs just fine?

When I first start my car, it starts right up, but if I don't let it warm up at least until the temperature gauge is at least halfway between C and H and try to put in in reverse or drive, it dies on me. It will not restart unless I wait at least 15 minutes, then it will start and then it runs just fine. I noticed while I am waiting those fifteen minutes that after about 13 minutes, it starts making the normal sounds of like the dinging when you put the keys in the ignition or the sound of the seatbelt safety reminder, etc. I then know that it is ready to try and restart and so I do and it starts and runs just fine, until it sits again for an hour or even longer, then I have to go through the whole process again. I was told it could be the fuel pump, so I have one on hand just in case. I took it to a garage yesterday and they replaced the IACS. It seemed to be running just fine all the rest of the day, but then this morning it did the same thing again. It was only in the single digits this morning, so I am hoping it was just because it was so cold outside. It has run just fine the rest of today. I was also told it could be in the ignition, maybe the switch or a relay in the ignition. I would gladly accept any help you can offer. I will send a donation to you if you give me your address. I do not have a credit or debit card to send one with. Thanks so much!
Kim