Welcome Deanna
.
"When I started the car and got out to check my back tires. I smelled gas."
This is quite normal as it runs rich right after starting and it takes a little longer for the idle to come down during winter. So don't wait for it to come down. As soon as the windshield it not foggy after starting, put it in gear and go.
As suggested previously, change the filter as it could be clogged up and make sure gas cap is closed with at least "three clicks". This is stated on the gas cap.
"I was told that it was my Air Filter."
I doubt it is your air filter but changing it is a good maintenance item.
"take off the positive battery cable thing for 10 min. This will release the settings that my car has made for itself"
This is to reset the PCM. An easier way is to open the Power Distribution Box (next to the battery) and pull out Fuses 4 and 11 for a minute or two.
"Doing this will make my car not idle like it is. And the gas smell will go away."
If resetting the PCM makes the smell go away, it means, the car was running on a limp mode before you reset it.
The next time you have the CEL come on, take it to Autozone or Advance Auto and have then scan it (free service). Make sure you note the Code number, then post it here for us to help you.
"I am assuming that this problem is the reason that my check engine light flashes on and off everynow and again when I am driving at speeds of 60+."
This could be misfirng but we won't know unless you have it scanned and post the codes.
"But I have other electical issues, that work on rare occasion. Dome light"
Probably due to winter, the spring does not work too well. Just push the switch by the door in and out a few times and see if that helps.
"light above door handle."
Probably broken. Don't worry about that.
"I am wondering if there not all connected by some type of wireing.
Am I correct?"
Could be but take care of the gas smell first.
Oh and do post your engine (6 cylinder Duratec or 4 cylinder Zetec). When you open the hood, you will see if it says Duratec or Zetec.
What year? Auto or Manual? And what mileage?