A rogue entering a fight late will only be missing out on potential assassination targets

How to deal with combatants entering a fight after it has already started

This part of your question is already addressed in another Q&A: How to handle initiative when a new force joins a combat in progress.

The answer is simply that when a new combatant enters the battle, they roll initiative like normal and go on that initiative.

This is also the same suggestion that Jeremy Crawford has:

Honestly, we never should have had spells tell you to roll initiative for creatures. The general rules just need a line that says, "If a creature joins a battle that's already in progress, roll initiative for the creature as normal."

When a combatant enters a battle late, they do not change who has already taken a turn or acted in the battle before their arrival.

Assassination effects

There are two effects that you can get from assassinate: advantage against enemies who act after you and automatic critical hits against surprised enemies. For different reasons, both effects only care about when the battle starts and not when the rogue enters it.

Advantage

You have advantage on attack rolls against any creature that hasn’t taken a turn in the combat yet.

This does not not say "taken a turn in the combat against the rogue". In fact, that statement doesn't even make any sense. You either have taken a turn in combat or you have not. There is nothing that measures how many turns you have taken in relation to a specific creature.

It is pretty clear that this ability is supposed to be used at the beginning of a fight, not at the beginning of the fight with respect to the rogue.

Jeremy Crawford has agreed with this:

Assassinate works only during the first round of a combat, unless a creature joins a fight later and has a lower initiative.

Note that by "a creature" he is referring to a potential target of assassinate. If an enemy joins the battle late, they will be having their first turn in combat and assassinate will work on them. If the rogue enters a battle after everyone has already taken their turns, then they do not get to assassinate anybody already in it.

Automatic critical hits

In addition, any hit you score against a creature that is surprised is a critical hit.

The rule on suprise says:

Any character or monster that doesn't notice a threat is surprised at the start of the encounter.

After the battle starts, there are no more surprised enemies. So, joining late also means no automatic critical hits.

Thus, a rogue entering a fight late will only be missing out on potential assassination targets.

The initiative rolls can be interpreted as a metaphor for other circumstances. So if the orc rolls high, imagine that as him turning his head ina lucky timeframe, just to see the attacker in the corner of his eye. But he is still surprised, so he does not get to act first.

To answer: the action that causes the surprise does count as a part of the round. The initiative is rolled when it is obvious that a combat should start, i.e. when one of the players or the DM decides to attack. Whether you roll it before or after determining surprise is of no consequence. But it is definitely rolled before the first attack (the back-stab in your case).

See e.g. this question. By the rules, it would go like this:

• Two opposing parties meet (Renee meets with the orc sentry) with intention of combat (Renee want to stabby-stabby).
• The DM decides which parties are surprised (the orc is, probably).
• Initiative is rolled. Now there are two possibilities – either Renee beats the orcs initiative or vice versa.

If Renee "goes" first: Renee uses her round to abuse the poor orc. Then the orc's round comes up, he cannot use any action or move on this turn and he couldn't have used any reactions up to the end of it (so for example if Renee moved away after the stab, she would get no attack of oportunity). From now on, combat goes as usual, next turn is Renee's.

If the orc "goes" first: The orc cannot do much on his round, since he is surprised. So his turn ends. From now on he can take reactions and normal actions on his following turns. Renee goes next and she spends her turn with the backstab (if she wants to). If Renee then wanted to move away from the orc, the orc could use the attack of opportunity on him. From now on, the combat goes as usual.

How this affects assassinate: If Renee beats the orc's initiative roll, she gets both the advantage (since she takes his first turn before the orc) and the crit (since the orc is still surprised)

If the orc goes first, Renee gets neither, since the orc is no longer surprised by the time Renee gets to her turn.