While I don't have the relevant book with me right now I remember the answer. It's a mix of both really. As Octavian was Caesars closest male relative and as he had been with Caesar on previous campaigns (in Spain for example), Caesar would have been well aware of the young man's witt and ability. This was certainly a contributing factor for choosing Octavian.

Caesar's choosing of Octavian as heir was a closely guarded secret and not many people knew (as was normal during this time). Octavian was in Greece at the time of Caesars death (dont remember where exactly) in preparation for Caesars planned Parthian campaign. So he did not know until after Caesars death. He arrived in Rome a few weeks after to claim what was his, to the dismay of Mark Antony who currently controlled Caesars assets.

EDIT: Octavian was actually in Apollonia Greece and not the Middle-East as I previously wrote.

Source: Antony and Cleopatra by Goldsworthy.

An important point is that with his testament Caesar made Octavian his heir in the traditional sense, that is, he was the one that inherited Caesar considerable wealth (and also replaced him in the center of his vast web of patronage).

But the testament did not automatically make Octavian the head of Caesar political faction, nor gave him in any way the enormous political power that Caesar had gained as dictator for life. In other words, as far as the romans understood those differences, Caesar named Octavian as his "private" heir, not his "political" heir.

Also, Caesar was still in good health, not planning to retire anytime soon, and certainly did not expect to be assassinated, so his naming Octavian should not be understood as a final settlement of the political situation after Caesar death.

It was Octavian who choose the immediately make a grab at political power at his very young age (he was eighteen years old, to become consul you should have been 40 or 42 at the time). That he was spectacularly successful it was certainly also because he made very good use of the wealth and the name recognition that he inherited, but it is mainly a tribute to his capabilities, his determination, and also to his luck (as when the two consuls with which he was in the field died of wounds leaving him as the sole commander of the senate army) and his complete absence of scruples.

When Octavian become Augustus and the first roman emperor he did assume a position and powers that Caesar could never have imagined, much less specified in his will.

Regarding your side question: Octavian did certainly often refer to Caesar. He made the Senate recognize Caesar as a divinity, and afterward referred often to himself as "Divi filius", that is "son of the deified one". It was an important part of his propaganda.

I've said all this before, notably in this thread, which links to another, much more useful thread that I wrote a while ago on a related question. The comments here so far are good, but none of them mention the most important aspect of Octavian's early career: he was not Caesar's original heir. Octavian entered Roman politics as a nobody from a fairly obscure family that happened to have a few connections but had no real ancestors of note. Octavian's father was a homo novus who died in 59 before he could stand for the consulate. Octavian had no background in politics and, at only 19 in 44, was totally inexperienced. Surely Caesar was impressed by him, or he wouldn't have made him his heir--Suetonius relates the famous story of Octavian's trek through the Pompeian lines in Spain, which supposedly left a fantastic impression on Caesar. But Caesar's original heir was almost certainly Antony. Caesar had two wills, and his original will was changed after Antony's disastrous governorship of Italy. It's almost certain that originally Antony was the heir in Caesar's will, and Calpurnia produced documents of Caesar's that claimed that he was making Antony his heir, only adding to the confusion when the new will was read aloud and Antony received almost nothing. The thread I linked to (errr...actually the thread linked to in the thread I linked to :3) relates all this in much greater detail, but Antony was not on very good terms with Caesar in the years leading up to his death, and Caesar was clearly very upset and disappointed by Antony's performance as governor of Italy, which was clearly his test-run to show that he was a capable administrator. Octavian was only named as heir as a sort of Plan-B, and undoubtedly Caesar exhausted a number of options before settling on such an unimportant figure.