Some notes...
Q: Can a Monster or Vehicle unit that is within Engagement Range of one or more enemy units be targeted with the Fire Overwatch Stratagem because of the Big Guns Never Tire rule?
A: No. Big Guns Never Tire allows a unit to shoot in its controlling player’s Shooting phase, but such rules do not take effect in other phases.
"... Deep Striking units have to come in more than 9" away from opposing units. So even if you come in 9.000001" away, an 8 leaves you at 1.000001", which is outside the 1" Engagement Range..." - User Tzelanit on Reddit.
So you need a 9+ after 9" Deepstrike for charging.
Fights First and Fight are basically two different phases.
In the Fights First phase - when both units have Fights First - the player who is not on turn starts the melee.
E.g. a Techmarine in a Landraider.
No. You can't use the ability from the embarked models because they're counted as not on the battlefield. You can use some ranged weapons if the Vehicle has Firing Deck X.
Your army can contain up to two units with each datasheet name – or up to four units with each datasheet name if it is a Battleline or Dedicated Transport unit.
Your army can contain up to one Titanic Character.