Questions about smoke

Don't use just a chimneys worth of charcoal every time, unless you want a hot fire spot right in the center. Have a bunch of lump in the fire box, and  spread the chimney's worth out for a broad, even fire, As it burns down, the chimney's worth will start the stuff below, and continue giving a pretty even tempertature across the whole cooking grill.

Whenever the dome is opened, the temp will spike. Not important.

Put the wood in at the start, on a mound of lump. Put in your chimney of burning lump. Let the visible smoke clear. The "good smoke" is mostly unseen. If the wood is in there, smoke flavor will happen.

Welcome @MichEgger‌. I light the egg similar to your method. I usually light enough lump in the chimney to replace what was used on the last cook. If I'm shooting for a 250° temp, I'll only use a half of chimney or less. One of these days I'll get a propane torch or weed burner. The cubes are hard for me to light when the wind blows, that's why I switched to the chimney. If the winds blowing I'll just start the chimney in the garage.

Anyway, as far as smoke goes, usually I mix chunks into the egg prior to dumping the burning lump into the egg. If I'm going to be cooking something like burgers or chicken and plan to do a dessert after the main dish, I'll wait right before the food goes on and use chips instead of chunks so the smoke will clear before starting the dessert. If I'm using chips, I try to toss onto the sides away from the fire to avoid burning them up to fast.