I coasted to a stop and smelled burning oil/rubber. I opened the hood and a little smoke came out. I'm not sure from where. I was so scared, I'm not sure if I tried to restart it. I had it towed back home the next day and on a whim I tried to start it. It started right up (stereo was turned off). No warning lights. I turned it right off. Fluid levels are good BTW.
Back story: I have a refurbished battery. I've noticed recently that when I run into town to do errands, after 3 or 4 quick stops, it will sometimes start sluggishly. Then a few days ago I hooked up my old subwoofer and 2 amplifiers. I was really pushing the volume and the subwoofer was working real hard.
So I'm hoping for bad Alternator or Battery. Could either have been the cause? Any thoughts?

 

This probably happened in reverse order --- died and then lights came on.

Any CEL codes. Have you ever done an ignition system tune up on the car?

 

Like Ron said, it is normal for the oil and battery lights to come on whenever the key is on but the engine is not turning.

Need to find where the smoke comes from, if it is still smoking.

 

Fluid levels WERE NOT OK. Very little coolant in the car. Happening in reverse order makes obvious sense once I think about it. I just found my "heater hose" split. The hose that goes from my header into the "thing" just in front of the firewall. The "thing" has a valve connected to a wire. When I move the valve, my slider on my dash moves from cool to heat.
Total new guy here if it wasn't obvious.

 

What engine and Civic trim (CX, DX, EX, Si)?

 

so I think what your saying it that one of your heater core hoses burst. That's good because that "smoke" you seen probably was coolant being burned after it started leaking. And also with the car starting to struggle to start after shutting off and turning back on 3-4 times may be a battery but if you want to know for sure go get your alternator tested, if its good I would start with a battery.

 

Replace that hose and any others that are the same age/condition as that one. Same with the battery, get a good name brand battery with a warranty not some refurbished cheapy.

 

It's a 1993 Honda Civic DX coupe with a 1.5L engine.
So I'm about to attach the new hose and top off my coolant. I'm thinking I may have a small leak in my heater system. Occasionally after I start the car I hear a momentary liquid sloshing sound coming from the passenger side footwell area.
This car had been sitting for about 3 years before getting it running again. Initially I blew a head gasket, replaced that and have put about 2K miles on it since.
It's obvious what I need to fix first but I'm curious to know why the engine completely died? It restarted and I shut it off quickly. Engine obviously didn't seize up on me.

 

The sloshing sound is air bubbles in the system. You need to follow a process to get the air out after re-filling. The 1993 should have an air bleeder on the fitting that connects the top radiator hose to the head.

 

Nice! I see the fitting. I'll google how to fix that. But if your responding so quickly, how do I do that?

 

So I just replaced the hose, filled up the radiator and reservoir. I ran it for 5 minutes and I got a few good burps out. I haven't ran it for the 15-20 minutes that you tube videos suggest. Everything sounds ok. It's running which is a huge relief. But I did see a bit of smoke coming from around the dip stick. Thats why I turned it off after 5. Thoughts on where the smoke is coming from?

 

^ Could be just sploshed coolant/water evaporating off. Keep an eye on your temp gauge.