Does Mercedes have a child company like Honda has Acura?

Or how Toyota has Lexus, if so what is it called

Daimler is the parent company. Smart car is another product line. Toyota is the parent company of Lexus and Subaru. But Mercedes is the "child company" for Daimler.

Chrysler was another child company of Daimler, until it (Chrysler) was sold to fiat (Hey Samsung, why does your phone refuse to capitalize some words even when you press shift?)

They have AMG, which is their high performance tuning division.

Not now. They DID attempt to sell their Maybach as a separate brand, but sales didn't justify the expense of maintaining a separate dealer network, so they moved the brand back into the M-B lineup. Mercedes also has AMG as their factory hot rod builder.

No