This is not a problem for most effects. Most of them last “until the end of your next turn”, which for effects that benefit something you want to do on your next turn is plenty long enough.
The only case where that produces counterintuitive results is when an effect disadvantages your opponent, as in your defensive spell example or a monk's Stunning Strike, because if your next turn comes before they next act then the effect is “wasted”.
This second class of effects is easy enough to identify on-the-fly though. Since you're already using a variant rule, it's easy (and totally true to the spirit of the rules as written) to house rule that such effects last “until the end the next round”, giving a full round plus a few initiative counts of effect, fully covering the opponent's next action.
The only drawback of that is that it might cover more than one of your opponent's next turns, if the initiative counts fall that way. Why do I say that's a drawback? Because the enemy can do the same to you, if they use such an effect against you and the dice go their way. However, that makes it pretty fair: the PCs might occasionally get the double-dip benefit, but so can the enemy. And the dice are likely to more often go in the heroes' favour, since that's the overall design of the rules.
Yes, 8 kobolds all going at once can be very swingy at low levels. To help this, don't roll for their damage - just use the average damage number.
I break monsters up into groups of three to five, to avoid this problem.
For example, you could have two groups of three kobolds. Roll initiative separately for each group.
That way you might get something like: PC1, then kobolds, then PC2, then PC3, then kobolds, then PC4.
For each combat, I write the combatants down on a scrap piece of paper, from first to last, along with initiative rolls. My minis are all coloured and numbered (OK, they are cardboard cutouts, but I'm calling them minis :-). So, I might have written down "21: Alice, 18: Kobolds red, blue, green, 16: Bob, 15: Eve, 14: Kobold Purple, Black, White, 9: Trent". I use this piece of paper to track the combat - HP, status effects and so on.
At higher levels there are abilities that really shine when players win initiative (Rogue Assassin archetype, for example). If monsters always go on initiative 10 then these abilities can become quite overpowering.
Other alternatives include the variant initiative rules in the DMG and "popcorn initative" as described by AngryDM (warning, bad language and strong attitude).
Best Answer
1. Should you roll initiative separately or together?
Making just one initiative roll for a whole group of monsters is normal play:
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:
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.