Failed attempts to kill important planetary figures will usually lead to at least some rebellion against the higher authorities. It was such an event that triggered the Vraks conflict after all.
A recently appointed charismatic Cardinal-Astral named Xaphen had been twisted to excess zealotry by one of his Deacons, who was subtly corrupted by Chaos. This Deacon encouraged the Cardinal-Astral to rouse the people into a war of faith to cleanse the sector of the heretical and unclean. This however is not something the clergy are allowed to do, as part of the Thorian Reformation, so they decided to start their Crusade on the Armoury World of Vraks, which was fortified and equipped and would serve as a formidable base.
Xaphen's oratory was enough to get the populace on his side save for his Sororitas bodyguard and the local Arbites forces, who plotted together to kill Xaphen during a speech. Unfortunately for them his Rosarius saved him from their sniper's shot and the Sisters and Arbites were killed to a man by the populace for their 'heresy'.
This gave Xaphen (who's paranoia was being fed by his Deacon) more room and evidence to claim all the people off-world were corrupt, that enemies surrounded Vraks on all sides, etcetera etcetera and make the Vraksians willing to fight the Imperial forces sent to retake the world.
Since most planetary leaders are demagogue's of greater or lesser skill, and the tiered nature of Imperial governance meaning most people only know a few levels of leadership above themselves to any real extent, a failed attempt to kill the leader of a planet easily lets them rile up anti-Imperial sentiment among the populace.
Depends on the circumstances. The Inquisition is very intentionally the opposite of a secret organization, and try to spread their reputation as far and wide as possible. Certainly there are many people who believe that Inquisitors are functionally unstoppable if they don't believe they're the hand of the God-Emperor.
Unless something has changed in Codex: Inquisition from all the older Inquisition material, they are very explicitly not an organisation that spreads it's reputation. It's rather explicitly stated in several books that the Inquisition is thought of as a myth by most of the population, usually works in the shadows (guys like Karamazov are exceptions to the norm) and swears most people who learn of them to secrecy on pain of death or forced recruitment.