Burnt valve would back fire and run rough, burn oil, wouldn't it? The vehicle isnt experiencing any of these.

If it isn't leaking it is burning in the engine. simple as that.

A common type of valve failure is valve burn, or burnt valves. Essentially this is caused by combustion gases escaping between the valve and valve seat when they are not sealing correctly. The hot combustion gases are forced past the valve which starts to burn away the edge of the valve, progressively getting worse the longer it is left without being rectified. Normally this type of failure affects the exhaust valves only, but it can also damage the intake valves.
A burnt valve will cause issues with your vehicle's performance and fuel consumption. Rough idle, reduced power, backfiring, and misfire are all symptoms of valve burn. If your engine is showing symptoms like these continued driving with a burnt valve will cause more damage to your engine, and will in turn cost more to repair in the long run.

Ticking may be valve lifter collapsed. Rough idle and oil burning could be burnt valve and poor maintenece. Run compression test to determine condition of rings and valves.

Gas in the oil can only get there threw the injector/s. One is stuck open. replace them all if it is worth the car. Back firing, I would check compression on the cylinders first, add a little oil first to all cylinders first, one or more cylinders are dry from the fuel, before putting any more money into it. Back firing threw the throttle body, could be timing or burnt valves.

Is there any white smoke? Could be your valve stem seals. But the Toyota Corolla in this year's is known for burning oil. If you are having to add more than 1 quart per month. You may have a problem. Check for the obvious. Milky oil. Idle it to op temp and smell exhaust for burnt oil smell. Hit the gas...see any white smoke? Open your radiator cap after running and smell for exhaust... If all is well then you have a normal running toyota corolla. These cars are champs. They just tend to burn a Lil oil here and there.

Experienced mechanic 10+ years

Your transmission is failing. Your clutches are burnt. Your check engine light should be on by now. By now you should be experiencing rough gear shifts amongst other things.

1 quart of oil in 1,000 Mi. isn't out of the range of normal for an older car. If it's used for mainly stop & go type driving [ city driving ] it's about normal. As things wear inside the engine some oil will find it's way into the combustion cylinders and burn with the fuel mixture, the catalitic converters will clean up the exhaust so you wouldn't see any smoke from this small amount being burnt.
A clogged air filter or faulty PCV valve will cause a higher rate of oil consumption, if those are good, I wouldn't worry too much about it. Just keep checking the level.

I would check the EGR valve first........Hope this helps.

check plugs -plug gap-vacuum leak-fuel mix-fuel pressure-timing-pcv valve-egr valve-air filter-oil sludge build up-problem in heads[valves adjustment-burnt-weak springs if only changed wires do a complete tune-up and do a compression check

Usually gas gets in the oil if you are having a hard time starting the vehicle, or turns over for a long period of time before firing. It floods the cylinders and bypasses the rings filling the oil with gas. You may want to change the oil and filter and see if this problem continues. Check your plugs, wires etc.. If it does then your rings may be shot not giving enough compression to the cylinder causing un burnt gas. Try a compression check and see if you have any bad cylinders. Depending on the vehicle 90-160 psi isnt uncommon. If there is a great differential between the cylinders you may have faulty head gasket, bad valves, or just worn rings.