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.
In order:
You can take the Ready action after you use a bonus action, sure. What really matters though is how you get your bonus action. If you just have a feature that lets you take your bonus action without any conditions then you can go ahead and do that. But some bonus actions have conditions that must be met in order for you to take them, like the one from wielding two weapons, as I will detail below.
Taking the Ready action does not count as a separate turn. What happens is, you decide on a triggering event, something you can perceive that you will react to in the future before your next turn. Then you pick the way you want to react to it. For example, you can take the Ready action to respond to an enemy approaching your companion and respond by throwing your dagger at him. Note: In order to actually respond to the trigger you described, you must use your reaction for the turn.
Two Weapon Fighting(PHB Page 195):
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand.
(Relevant Portion)
So from here we can see that, in order to get your off-hand attack with your bonus action, you have to have already taken the Attack action. This eats up your action for the turn, so you will not be able to take the Ready action like this, unfortunately.
What this ultimately means is that you wouldn't have a way to throw a dagger as a bonus action (unless you found a way in-game, like if your DM gave you a magic item). If you could do it though, you would be able to make both attacks with Sneak Attack, as RogueBurger points out in his answer with a reference to a Jeremy Crawford tweet.
Best Answer
That sounds like a reasonable use of Ready.
Here's the relevant rules text (my emphasis):
A character can (usually, see below) perceive when another character finishes casting a spell, or when they signal that they have done so, such as by shouting "OK, go". Characters can use Ready to coordinate their actions provided they give each other that type of perceptible cue. Guidance specifically requires that the caster touch the (willing) target, so that is also clearly a perceivable circumstance.
How perceptible a spell is when being cast depends on its components, of course, so in oppositional scenarios, where a character wants to Ready with an enemy spellcaster's spell as the trigger, the DM will need to account for how well the character can see or hear the caster, whether the spell has verbal, somatic, or material components, whether modifications to those components are in use (via subtle spell metamagic or other effects), and so on.
In short, the trigger has to be something the Ready-ing character is aware of, but not necessarily a physical event.