I believe you are near Sacramento which can have cooler temps at night. It could very well just be a choke issue, sounds like a mechanical choke right? When it's warmed up if the choke is still closed you could smell rich fumes smell and it won't run right. Then it sounds like you got the choke to open up and it ran fine, but then wouldn't cold start because the choke was open now.
Go to an auto parts store and purchase an inline spark tester, they are inexpensive and easy to use but more importantly it will tell you whether it's spark or fuel, to me it sounds like a fuel issue but at least with the tester we can be more certain.
The carburetor should be a Rochester Quadrajet if it's still the stock carb. I personally love Rochester carbs for daily drivers, they just work great. But, I have seen far too many mechanics not understand how Rochester carbs work since they are somewhat complex and if not working properly will give problems. I believe in 1987 the carb might have some emissions controls which cold be part of your problems.
Personally if it were my car and I didn't possess the knowledge and skills to work on a quadrajet carb I'd just remove it and all the emissions crap and install an Edelbrock 500 cfm 4 barrel. The reason for the Edelbrock is you have a spread bore intake and you won't be taking advantage of the tuning advantages of a Holley. Edelbrock's are also easier if you do want to tune it and don't require removal and disassembly to change jets etc... It sounds like you probably won't be doing that sort of stuff and just want the car to run right and not go out racing or to the track. You can maybe go up to a 600 cfm if you are redlining over 6000 rpm, but for 100% volumetric efficiency and 5000 rpm limit 500 cfm is a good choice and better on fuel economy. I think the stock Rochester equipped 305 is only rated at around 180hp.
The other option is remove the Rochester and ship it to a professional to go through it. But then again you could probably have an Edelbrock at your doorstep in a few days and have it installed in one day and driving the next.
I had a customer recently that had a Pontiac 305 and Rochester that needed to be gone through. Instead of having me rebuild the carb he purchased a Holley double pumper which are not the easiest to setup for street use especially with a cammed and overbuilt engine. It took quite a bit of time to get his fuel to air ratios proper during all driving conditions with a mechanical secondary carb. Personally I don't see why mechanical secondaries are so popular with street cars. If you do go Holley, which is also a great carb make sure you get a vacuum secondary model and a spacer that fits the spread bore intake to the square bore Holley. Right out of the box a vacuum secondary Holley should work just as good as an Edelbrock but people I talk to find Edelbrock easier for a non-carb mechanic car enthusiast to modify. Edelbrocks are basically copies of Carters using metering rods.
