When I crank the car white smoke out the exhaust

That's a sign of burning engine coolant due to a leaking cylinder head gasket. The level in the coolant reservoir will go down over time.

Nope. We don't get involved with costs here because there's way too many variables. We don't know the labor rates in your area or the cost of parts. Also, BMW is the one manufacturer that is real miserable to deal with. They do not release service information or paint codes. Everything done by independent repair shops has to be learned by trial and error. BMW has a very selfish and greedy attitude that does not benefit the car owners.

On most other car brands a coolant leak into the engine is caused by a corroded cylinder head gasket about 99 percent of the time, but it is always possible to find the cause is a crack in the head. While that is rare, I don't know the history for BMW to know if it's any better or worse than for other brands, but there is no way to know until the heads are removed and inspected. Most repair shops will send the heads to an engine machine shop to have them checked for cracks and for flatness. If they are warped just a couple thousandths of an inch the new gaskets may not seal. Some heads can be resurfaced to make them flat but if an overhead camshaft is used the heads have to be heated in an oven and straightened, other wise that warpage will cause the rotating camshaft to bind and possibly break. Many cracks can't be seen without using a special dye. Aluminum heads can also be checked with a large electromagnet and magnetic powder. Repair shops don't typically have that specialized equipment. That's why they farm the heads out to a specialty shop.

The best your mechanic can do is prepare you with the worst-case estimate if two new heads are needed. He may also present you with a list of additional parts. For example, there's two timing chains used in your engine. They typically don't break but they will stretch causing reduced power at high speeds, and increased power at low speeds. You may not even notice that but given the mileage a conscientious mechanic will recommend replacing them as long as they are already removed to remove the cylinder heads. Some engines require new cylinder head bolts each time they're removed. They stretch when tightened and they only do that once. The second time they will not develop the needed clamping forces. Those are called "torque-to-yield" bolts.

Jobs like this can be hard to estimate if they haven't done one before, and when they don't know yet what is needed. They will have to wait until they hear from the engine shop as to the need for new head bolts and other parts. The engine shop may even have a kit already put together for your engine. Those typically save a little money over buying each part one at a time, but you may get more parts than what are needed. Usually they have extra gaskets or rubber seals but there won't be expensive parts that aren't needed.

I should point out too that all of this is based on your description of the symptom. Your mechanic isn't going to replace the head gaskets based on my diagnosis. There are two tests that can confirm a leaking head gasket. One involves testing the air coming out of radiator, and one involves adding dye to the coolant. After driving a while you search with a black light, particularly in this case, at the tail pipe. The dye will show up as a bright yellow stain. If you see it on the inside of the tail pipe, it got there due to the leaking head gasket or cracked head.

Nope. They just measure the amount of unburned oxygen in the exhaust gas. The Engine Computer fine tunes fuel metering based on those readings.