To address your questions in order,
First, stereochemistry is not a property inherent in reactions, it is a property of molecules. All molecules (not just chiral ones) have stereochemistry. However, in cases without a chiral center, we choose not to draw the molecule such that stereochemistry is depicted (we draw it without wedges and dashes). So, it doesn't matter how you react a molecule - it has stereochemistry. However, you only have to draw wedges and dashed is there is a chiral center.
A racemic mixture is created when the products of a reaction produce EQUAL left and right hand (d/l) enantiomers at a chiral center. Diastereomers, however, are stereoisomers that aren't enantiomers. For instance, if you have a molecule with multiple chiral centers, and you perform a reaction that reacts at and inverts one chiral center, do you get a racemic mixture? No, you don't, because EVERY chiral center is not inverted. Instead, you get a diastereomer.
In the example you presented, the H can add to either carbon 2 or carbon 3 because the molecule is a meso compound (symmetrical). Also, there is no opportunity for hydride shifts to occur, so you're good on that front.
The reason halogenation of alkenes leads to an anti configuration is because that is the configuration with the least amount of steric hindrance (strain) between the two halogens. The point is, if halogenation produces two adjacent chiral centers, then we well indicate the chirality (wedges/dashes). If it doesn't produce a chiral center, then the chirality doesn't have to be indicated, but know that the two halogens will have a configuration that minimizes the interactions between them.
Hopefully, I was able to shed light on some of your questions.