First of you should know that Edelweiss is a tricky scenario to get a BV in.
But to answer your questions:
The odds of getting an overrun are shown in the combat-calculation window. Anything less than 80% is a no-go if you really need an overrun. And RNG plays a roll in every calculation in this game - the odds and percentages shown are just that: odds and percentages. Don't become a victim of the gamblers fallacy
When you achieve an overrun you can use that unit to attack again - or move if it has not moved into an enemy ZoC. However each successfull overrun also surpresses one step of your unit so you cannot keep going forever.
The premise for getting an overrun is simply to have a much more powerfull unit attacking a weaker unit. Fully surpressed units are generally always overrun.
Overrun is key to many (if not most) scenarios. It gives you that edge you need to get through an enemy that outnumbers you. Sometimes its a good idea to weaken a unit with infantry to allow the more powerfull armour/mech/mot units to overrun and then breakthrough.
Veterancy is gained by killing enemy steps:
10 steps: Veteran
25 steps: Elite
Be carefull when reinforcing your veteran units. A veteran units needs an additional 5 kills to allow it to be reinforced with a single step without regressing to a normal unit. This number is 10 for an elite unit.
Hope this helps