1998 Daewoo Lanos White smoke on start up

White smoke is a sign of burning coolant due to a leaking cylinder head gasket. That has nothing to do with your climate. The clue is the level will be going down in the coolant reservoir.

What would be the fix for this. A new head gasket?

And also the tappet sound? Should I replace the valve stems?

Or just get a full diagnosis

Thanks

You need someone to listen to the noise to make a diagnosis. Valves don't make noise, but if the oil pressure is low, the lifters wont pump up to take up the slack in the rest of the valve train, and that will cause a rattle like a typewriter.

To replace a head gasket requires removing the head. That's a pretty involved job.

Thanks for your input. Do you think its wise to replace the head gasket? I also know the timing belt is way overdue. I also need a new strut, and the windscreen is etched from someone too lazy to replace the wiper rubbers.

Not sure if this all should be done, or just get a more reliable 2nd hand car.

You didn't say if you're losing coolant. There's a chemical test that can be done at the radiator to test for a leaking head gasket. If that comes up negative, a second test involving adding dye to the coolant will show if the coolant is going into the exhaust system. If a head gasket is leaking, there's nothing to be gained by ignoring it. The leak will continue to get worse, and will eventually lead to an overheated engine and more problems.

According to my reference material, the 1.5L engine is not an "interference" engine, so if the timing belt breaks, you'll coast to the side of the road, but no other serious engine damage will result. That reduces the urgency of replacing the timing belt, but if that service is needed, the belt has to come off when the head gasket is replaced, so you're better off doing both repairs at the same time.

Struts wear out on any car, so that is nothing to consider when deciding to fix or trade a car. They're a normal maintenance item.

Thanks for the valuable insight