Short answer: Yes, you can take a reaction before your first turn in combat.
Reactions and bonus actions aren't ever something you “have”, they're something you do. And there is a limit on the number of times in a round that you can do them.
As a consequence, it doesn't make sense to ask if you “have” either one when you are surprised. What matters is whether you can do things that count as a reaction or as a bonus action. The only reason you wouldn't be able to take a reaction before your first turn is if you're surprised (both quotes from PHB page 189):
Surprise
If you’re surprised, you can’t move or take an action on your first turn of the combat, and you can’t take a reaction until that turn ends.
Bonus Actions
[…] anything that deprives you of your ability to take actions also prevents you from taking a bonus action.
So since you're not surprised in this scenario, you can still take reactions.
Also note that surprise also doesn't occur just because you lose initiative. To be surprised:
character or monster that doesn’t notice a threat is surprised at the start of the encounter.
If you're used to earlier editions of D&D that have a “flat-footed” rule that make you vulnerable in a surprised-like way until your first-round initiative comes up, that might be the source of your assumption that someone can't act in 5e until the first time they have a turn in a combat. There is no similar rule in D&D 5e: if you're unsurprised, you are no more vulnerable before your first turn than after.
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
A houserule I use occasionally is to allow creatures (not players) to voluntarily drop their place in the initiative order, a combination of rolling one initiative for a whole group and readying actions in order to act simultaneously.
Roll all the initiatives separately, then when it gets to a creature you want to be 'in the group' have them ready an action to take with their comrade. When it reaches their comrade's turn in the initiative order the readied action triggers and you simultaneously drop that creature's place in the initiative order.
Mechanically this is the same as rolling initiative for all the creatures in a group but then giving the whole group the lowest initiative rolled as a nod to how hard it can be to 'form up' and organise into a unified fighting force. Thematically it's nice to describe the soldiers bringing their pikes to bear/the thieves giving each other a subtle nod/the wizards holding out their arms in preparation. Doing this description in different places in the initiative order can give the players a bit of a heads up that when their enemy does move it's likely going to hurt so they can prepare defences or try to disrupt the 'formation', and the payoff if a Paladin PC notices all the enemy archers are readying to fire at the frail old wizard PC just in time to stick himself in the way is very satisfying.