There are 2 definitions of "round" in 5e. There is the definition of "round" that you quote, which is from initiative position intMax to initiative position 0 (or negative if you manage that some how).

Then there is the definition used here for readied actions. This is the same definition used in "once per round" effects such as certain powers. This definition begins at the beginning of your turn and ends at the beginning of your next turn.

Because (unlike in 4e), readying an action and activating it, doesn't move your initiative order position, and because it matches the readied action refresh timer (which happens to use the same 1/round definition of round), readied actions can be used at any point in either the current round or the next round prior to your turn.

Here's how reactions refresh:

When you take a reaction, you can’t take another one until the start of your next turn. If the reaction interrupts another creature’s turn, that creature can continue its turn right after the reaction. (Player's Basic p70)

This is a large part of the support that leads me (and many others) to believe that readied actions can roll up to your next turn.

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.