Wow! Can't believe I didn't see these until now!
Caesar is offed by one of his legates, and Pompey is secure - what does he do?
Well, in this situation, it depends on when Caesar is killed.
1:
before the Battle of Alesia: Labienus (or whomever takes command afterwards) would probably be forced to retreat from Gaul, and the returning army would demand payment from the senate, so the civil war mights start without a leader (or perhaps Crassus would assume command of Caesar's forces if the POD is before Carrhae).
2:
between Alesia and the Civil War: Pompey would likely act quickly and send his senatorial allies to take the Gallic commands and consolidate power. Pompey remains the strongest man in Rome, he gradually alienates the senate, and they flock to the next charismatic upstart who can challenge Pompey, the cycle continues
3:
during the Civil War, before the Battle of Dyrrhachium: Caesar's death leaves a power vacuum in his faction, and one of his commanders acts swiftly to consolidate power. The likeliest candidate is Antony, but there are also men like Decimus Brutus, Marcus Lepidus, Gaius Curio, or Domitius Calvinus who might each gain supremacy within the first year of the Civil War. Whoever emerges from the conflict is gonna be facing an uphill battle, since Pompey will have gained a lot of strength in Greece in the meantime, and with their superior numbers, an invasion of Italy may be possible around 46-45 BCE.
4:
during the Civil War, before the Battle of Pharsalus: Caesar was in dire straights after Dyrrhachium, and one of his troops actually tried to kill him IOTL during this battle, so imho, this is the likeliest time for the assassination to happen. If he died at this battle, his army would likely surrender, facing an overwhelming numerical disparity and very poor supply lines. ITTL, Antony would attempt to rally a defense of Italy, but with Caesar dead, Pompey's victory would be pretty certain (similar to how Cinna’s assassination made Sulla's victory much likelier)
Latin Right scenario sounds really intriguing. If you don't end up doing it could you at least share some ideas of how you imagine it'll go?
Probably it would involve a lot more development and colonization in Sicily, Sardinia, and Corsica (which remained rural backwaters for pretty much all of Roman history, despite being the first provinces ever incorporated). This would mean the cultural patterns, military culture, and political class would be more fully incorporated into the Republic. This would firstly, set a precedent for the incorporation of conquered peoples as socii. This would make the potential recruiting base for the legions much much larger, but also bring in a larger pool of wealthy socii into the political class of the Republic. It probably means a much earlier social war, possibly at the same time as the Punic Wars, leading to a much different expansionary history of the Republic. The Republic would probably be much slower to expand, since the incorporation of socii would make it more difficult to politically digest these new regions. Spain would probably be completely off the table for annexation under this system, and Rome would probably stick to annexing coastal settlements and not venture very far inland, at least until after the conflict of citizens vs socii is resolved.
Could the Ostro-Gothic Kingdom in Italy be seen as "Roman"?
This is a good one imo, but it would require Odoacer to put an actual Roman citizen on the throne. Some senator would suffice, but he would probably pick a child, since child emperors can't do much by definition. In this view, as long as he keeps his army happy, he'd be politically invincible from threats within Italy, and he'd probably spend his reign fighting a three-front war against the Visigoths, Vandals, and ERE. It's unlikely he'd last too long under those circumstances, but he may be surprisingly resilient (as happened IOTL).