1. Should you roll initiative separately or together?
Making just one initiative roll for a whole group of monsters is normal play:
Initiative determines the order of turns during combat. When combat starts, every participant makes a Dexterity check to determine their place in the initiative order. The DM makes one roll for an entire group of identical creatures, so each member of the group acts at the same time. (PHB, p. 189)
Making one roll for a group is often advisable, especially for new DM's as it both speeds up and simplifies combat.
(However, in large groups this can sometimes cause issues.)
2. Always roll the monsters attacks separately.
According to the Order of Combat:
During a round, each participant in a battle takes a turn. The order of turns is determined at the beginning of a combat encounter, when everyone rolls initiative. (PHB, p. 189)
So, every kobold will take a turn each round. And, they could all choose to use their action that turn to attack - so they could all attack each round.
However, there's no guarantee that they would do so, as there are plenty of ways an NPC could spend their action other than attacking (Dashing or Helping, for instance)
If they do attack, they can definitely attack different people. If only one kobold took an attack each round then you'd really only be fighting one kobold at a time three times in a row (a much easier fight), rather than three kobolds at once.
When multiple creatures are attacking, even with the same target, all attacks should be rolled separately (though the separate dice could be rolled simultaneously). This prevents damage from being too swingy.
If either everyone hits or everyone misses in any given combat round then it becomes harder for players to gauge how hard a fight really is, and more difficult to prevent PCs from being knocked out through careful healing.