[iceminion]

I've been daily driving my DSM's for about ten years now, so I would say that I have some experience with this.

First off, for your water pump:
-I always keep the belt as lose as possible.
-I understand that there is an "official" way to check tension, however I have chosen a better way.
-losen the lock bolt on the alternator tensioner.
-start out lose, it should squeal on startup.
-turn car off, tighten tensioner bolt one turn, try again
-if squeals, tighten one turn, try again.
-tighten bracket tensioner "lock" bolt(short).
-reinstall AC fan.
-further adjustments may be necessary, but when using used pumps, used belts, and dirty pulley teeth, this is the best method.
-overtightening the belt causes bearing failure, remember this.

OK, now on to why your car is taking a while to turn on:

My gut feeling that there are a lot of things contributing to this:

First off you have two separate issues that are linked.

however you must recognize that there are two separate issues or what I say will not make sense.

Issue #1 - delayed start.
Issue #2 - dies once running until cylinders heat up.

To resolve both these issues, treat for the following:

-Poor injector spray pattern (have lots of supporting evidence, build your own injector testing station, easy to do, will share info by request)
-Deleted fuel pressure assist solenoid (we all delete this, we all don't care, but it does affect startup in the cold, put it back.
-Verify that your fuel pressure remains between starts, if the fuel pressure valve in the fuel pump has been damaged/removed, all the pressure bleeds back into the tank and you'll have air in your fuel rail.
-Your CAS is 180 degrees off, trust me, it is, set engine to TDC and set CAS to alignment notches and re-install.
-throw away your blacktop CAS, buy a greentop CAS.
-re-gasket your CAS, believe it or not some CAS sensors get damaged by oil creeping into the chamber, take the security screws out (have tips for that) replace with torx security screws, clean up sensor, use "the right stuff" for a replacement gasket. (do NOT touch the stuff on the inside)
-Your ECU is damaged, come to terms with it, its true, talk to keydiver, you need to replace IC104 (thats how it's labeled on your ecu.
-I do have good ecu's from cars that start instantly.
-I also have good, "new" gen (black) IAC's if you need a known good one.

Let me make a transition here to ECU repair/maintence/testing because it is part of this. Now test your idle control system:

-Adjust BISS to idle 750 RPM when hot (if your vehicle does not stay hot, this will not work because your car is in warm up mode, im gonna post up about thermostats in another thread, read it).
-Drive at over 30+ MPH (to hit the cruise control VSS switch engagement point)
-Come to a complete stop in "N" or "center" on the shifter, only use brake pedal.
-During the slowdown, your RPM NEEDS to be 900-1200 RPM (IF IT IS NOT YOUR SH1T IS BROKE, ASK FOR HELP)
-Once your car comes to a complete stop, the idle NEEDS to drop back down to 750 RPM(IF IT IS NOT YOUR SH1T IS BROKE, ASK FOR HELP)
-Rince and repeat, if the car does not do this reliably there is something wrong with your idle control system.

Now that your idle control system is working, your stock 93 turbo 7-bolt engine will start up almost instantly, idle better, warm up faster and just work better.
Dirty/damaged CAS sensors also contribute to 2G's random misfire issue, I've done 6 2G 6-bolt swaps, all work great with the stock ECU, no misfires at all, thats why its a "random misfire" nobody knows why some cars just work great after the swap, I do.

I have several examples in my driveway that are proof that DSM's turn on almost immediately in case you have any hint of disbelief.

If this is a little dis-organized, I apologize, I will revise it when it isn't 3AM.
If I come off as an asshole, sorry, kinda hard to give technical support for your car and for your emotions at the same time, read around the things I say that piss you off, absorb the information, move on.