Caesar was deeply involved, but probably not as a conspirator. In the years immediately following the Catilinarian Conspiracy it was standard invective to claim that one's opponent had been a Catilinarian who had escaped justice--Cicero accuses just about everyone of this at some point. Caesar's political career overlapped with that of Catiline at several points, however. Both Caesar and Crassus had openly backed Catiline's canvass for the consulship of 63, and they were left in a somewhat embarrassing position, since even though Gaius Antonius had been elected they hadn't managed to beat out Cicero entirely. Caesar had also presided over the court that had acquitted Catiline of murder in 63, and though it had of course been the jury that had voted not to condemn him, it's not hard to imagine that Caesar's influence might have had something to do with it. As early as 64 Cicero was making claims that Crassus and Caesar were involved in some sort of plot with Catiline and Antonius, if we are to trust Asconius' commentary on the In Toga Candida. But while Caesar and Crassus were supporters of Catiline's political career, there doesn't really seem to be much of a reason to suppose that they were actually involved in the conspiracy. Both Sallust and Plutarch insist that Catulus' and Piso's accusation that Caesar had been involved in the conspiracy was due to personal enmity, either because of Caesar's election as pontifex maximus or his prosecution of Piso for executing a Transpadane without trial. They may have been involved in the so-called First Catilinarian Conspiracy, the plot to murder the consuls of 65, but Asconius says that Cicero placed the blame for that one solely on Crassus, and Plutarch mentions (in a passage of rather suspect truth) only Caesar and Crassus as having plotted in 65, with no word of Catiline.

Certainly during the conspiracy Caesar appears to have done everything in his power to distance himself from the Catilinarians. Suetonius relates that when Novius Niger accused Caesar in 62 of involvement with the Catilinarians (on the basis of testimony from Lucius Vettius, who had tipped Cicero off about the conspiracy, and Quintus Curius) Caesar pointed to Cicero's testimony that he had in fact informed against Catiline to Cicero--the letter to Catiline from Caesar that Vettius claimed he had (but apparently never produced) was rendered irrelevant, and if it existed in the first place there's no reason to suppose it was connected to the conspiracy at all, given Caesar's support of Catiline's bid for the consulship. Likewise, Crassus also distanced himself from Catiline during the plot. Sallust says that Crassus was accused of telling Catiline to not be bothered by the arrest of the conspirators, but Tarquinius' testimony was ruled by Cicero as false. According to Sallust the accusation was trumped up either by Publius Autronius to get Crassus to join forces with the conspirators and defend them all, or else by Cicero himself, to prevent Crassus from defending the conspirators and force him to distance himself from them--Sallust claims that he personally heard Crassus offer this possibility, claiming that Cicero had insulted him (tantam illam contumeliam sibi ab Cicerone inpositam). Both Caesar and Crassus were responsible for the imprisonment of two of the conspirators, Caesar holding Statilius and Crassus holding Gabinius. Sallust, Appian, and Plutarch claim that during the actual investigation Caesar was placed under suspicion, but that Cicero opted not to charge him. Their reasons are pretty different, though. Appian thinks that Cicero was afraid of Caesar's popularity with the people, although according to Appian it was Cato who openly accused Caesar of being one of the conspirators after he suggested that the Catilinarians not be put to death. This sounds like standard rhetoric on the part of Cato, and it doesn't line up very well with what Sallust says, namely that Cicero thought the accusations of Catulus and Piso were absurd. According to Sallust Catulus and Piso spread rumors about Caesar after Cicero refused to charge him and the equites that had gathered around the Temple of Concord during Cicero's oration there believed these rumors and threatened Caesar with swords. Plutarch thinks that Cicero didn't have enough evidence to charge Caesar, though in the life of Caesar he says that it's not clear whether Caesar actually had anything to do with this. Plutarch also says that some people say that Cicero feared Caesar's connections, but he doesn't seem to find it very convincing, and given Cicero's reputation among a number of very influential circles neither do I.

So at most we can say that Caesar was accused of having been involved in the conspiracy, though lots of political enemies in later years would be as well. The case against Caesar seems to mainly have rested on his and Crassus' support for Catiline against Cicero's bid for the consulship, and on Caesar's proposal not to execute the Catilinarians without trial. There really doesn't seem to be enough to link Caesar to the conspiracy, though, and our texts are unanimously skeptical of the idea or totally dismissive of it. Caesar personally may have sympathized with the Catilinarians, although quite a lot of people who in later years would be known for similar policies and would be connected in invective to the Catilinarians were in 63 very much opposed to them (e.g. Publius Clodius, who was certainly no supporter of the conspiracy, though Cicero made him out to be one later).

There is really no way to know. He was certainly never close to being indicted much less executed. Suspicion may have started because he led a push to exile the conspirators instead of execute them (the conspirators were not given a real trial so execution was a pretty big step). He gave a convincing speech but the pro-execution argument prevailed. When he was consul he was very untraditional and carried through a few major reforms, which could theoretically be seen as Catilinarian. Finally, he allowed (perhaps encouraged) Clodius to become a plebian and thus run a tirade against Cicero ending with Cicero's banishment from Rome for a year. All these things combined certainly made him seem pro-Catiline.

The big issue is that ancient authors don't even agree how Caesar got wrapped up in everything. Sallust claims that Caesar was accused by Catulus and Piso who were both trying to get back at him for slights (he denounced Piso at a trial and beat out Catulus for pontifex maximus). Plutarch gives a similar story but he claims Piso was also jealous because Caesar was pon. max. not for his personal reason. Finally, Suetonius omits the two entirely and claims Caesar was accused by someone just trying to get reward money. Not only were these historians confused about how Caesar was implicated, they were unconvinced.