BMW Repair: 87 535i Wont STAY Started If Cold, couple factors, bmw owner

Question
My 87' 535i runs like a champ always, UNLESS IT'S COLD, then it won't even start. It was doing this a couple months ago, then it just stopped, and was starting fine, EVEN WHEN IT WAS COLD OUT. (I mean cold for Los Angeles, which is below 50.) If it's cold out, it turns over fine, but just won't start. Then sometimes it starts RIGHT UP, but only stays started for one whole second before dying! I wait a minute or two, it turns over and starts RIGHT UP, then dies immediately. I don't need to press on the accelerator when starting since it's fuel injection, so IT IS NOT FLOODED. I can assure you it's not flooded, or it would not turn over and start so quickly EVERY TIME like that. I sat in my car recently after work and played that stupid game for 45 minutes. IT WOULD NOT STAY STARTED. I never know when the hell it's going to do this. It's a little maddening now. I had to leave it at work and take a damn bus home. (No BMW owner should be taking a bus home from work.)
Here's a couple factors which may play into it. The temperature gauge in the car HAS NEVER WORKED. This means that the wire from the sending unit is shorted out or corroded somewhere. Could the fact that the gauge doesn't work, mean that the computer doesn't know what the engine temperature is, causing this reaction? Just a thought. Could that be effecting the DME chip somehow? If I get that temp gauge fixed, do you think it might solve this problem?
Is it something totally different?
Someone suggested the vacuum pressure in the engine. It's not that. Someone else said a new DME chip. Those are EXPENSIVE (Rebuilt on pelicanparts.com is $497.00.). It's got to be something else.
Someone mentioned maybe a new cold start valve? We're having a cold snap out here in California, and it's below 50 degrees and will not start at all right now. It started and ran fine yesterday. If it is the cold start valve, do I also need a new thermo-time switch? It's making my life miserable, and I really want to get to the bottom of this.

Thank you.

Tom B.

Answer
Aa few thoughts:

Your temp gauge not working won't impact the car....IF it's the GAUGE that's not working.  If it's the sensor that's bad, and that same sensor talks to the DME, then that definitely could be the problem.


If this were an older car, it would sound like it's a choke issue.

I wouldn't think the DME if cold starting is the only issue.  Not sure what the cold start valve is on this model, but that would sound logical.

Good luck!