Rough Start And High Idle

Hey guys, I just got a 2000 GT with a 4.6 and a 5 speed manual transmission. Whenever I start the car it will start, immediately die, and after it dies it will backfire. It will generally do that one time and then it will start right up and run fine. Also, the idle is a little off I think. When its sitting it will idle just fine. If I drive it and then come to a stop it will idle at about 1500 for a little bit and then eventually come back down to normal (around 850-1000) Is this normal or is there something going on there?

I have read that the cam position sensor can cause the first issue but is that related to my idle problem or could that be something else? IAC maybe? I am pretty handy with cars but most of the work I have done has been on a Wrangler which is like the polar opposite of the mustang.

Any help you guys could give is greatly appreciate. Also, if you need more info just let me know because this is my first mustang so I am completely new to all of this.

One other thing I forgot to mention is that when driving if I get up to about 3k rpm's the battery light will come on but as soon as I shift the light goes away. I know the battery itself is good because I just put a brand ew one in about 2 days ago. So the alternator could be the cause of the starting issue and potential vacuum leak for the idle problem?

Alright so I got out there for a little while today and the vacuum hose to the pcv is all dry rotted and cracking and leaking. So I am going to change that and look into getting a new alternator.