If an enemy ship is close to 2 objective tokens in this mission...

1. Can I reroll 4 dice from one big volley?
2. Can I reroll 2 dice twice from one big volley?

Just two dice in total, unfortunately (FAQ p. 10). As if this one needed to be reined in...

Minefield markers are the only ones that can trigger together I think

As @Villakarvarousku said, you can only reroll 2 dice. The effect on Targeting Beacons uses the word "while" which is explicitly defined in the RRG that those effects can be resolved only once during the specified event and cannot occur again during that instance of the event.

In Minefields for example multiple tokens can trigger in the same manouver because it says "If a ship ends its movement..." instead of " When a ship ends its movement...".