We know that without the union of Sarah Connor and Kyle Reese, John Connor would not have been born, and we know that without the remnants of the T-800, Skynet would not have been developed. Is it ever explained in novelizations, deleted scenes or other official sources that Arnie's mission was to prevent John Connor's existence, but regardless of the success of the T-800, Skynet's ultimate goal was to spawn itself?
No. The goal was only to kill Sarah Connor.
We can deduce this from the behaviour of the T-800. If its goal was to advance AI research and spawn Skynet, it could have taken some actions to advance this goal. It could have found a research group and given them ideas, schematics, even samples of its own hardware. In fact, there's no reason why it couldn't swallow a spare CPU, regurgitate it in the past, and hand it over to the researchers at Xerox Parc (or some other 1980s cutting-edge facility).
The T-800 isn't in any particular hurry. It could do all of these things before it started hunting Sarah, but it doesn't. Instead, it relentlessly chases after Sarah, putting itself at non-trivial risk of being destroyed by her and Reese, or by the authorities. (The police proved rather ineffectual, but if they'd realised what they were dealing with they could have called out the National Guard with heavier weapons.)
It's sheer fluke that the T-800's CPU survived being partially crushed in the hydraulic press, and then was picked up by the founders of Cyberdyne Systems. It appears all but certain the version of Skynet which sent the T-800 to 1984 did not plan for these events.
No, in the original timeline Skynet had been developed without of any information from the T-800. Skynet sent back the T-800 because preventing John Connor's birth was the only way it could avoid losing the war.
From the Terminator script, Kyle Reese explains that Skynet was developed as a computer defense system in the original timeline (which did not include any information left behind by the destroyed T-800):
Silberman: And who was the enemy?
Reese: SKYNET. A computer defense system built for SAC-NORAD by Cyber Dynamics1.
Reese also explains why Skynet was forced to prevent John Connor's birth rather than simply killing him during the war:
...it had no choice. The defensive grid was smashed. We'd taken the mainframes... We'd won. Taking out Connor then would make no difference. Skynet had to wipe out his entire existence. We captured the lab complex. Found the...whatever it was called...the time-displacement equipment. The Terminator had already gone through.
From Skynet's point of view, it did not need to ensure its creation because it was going to be developed as a defense computer whether or not its human developers could take advantage of any information from the T-800 (it was developed during the Cold War, so humans already had plenty of incentive to develop a defense computer like Skynet anyway). Its paramount objective to ensure its continued existence was to kill John Connor, and as Reese pointed out it had to kill him in the past (1984). Otherwise it would certainly be destroyed by the Resistance in the future.
And, as also pointed out, the T-800's behavior gives no indication that Skynet intended for it to give away information in order to aid its development.
1 This is from a draft script, and it's not clear if "Cyber Dynamics" is meant to refer to the Cyberdyne we see in Terminator 2.
This is specifically addressed in the film's official novelisation.
After killing Sarah Connor, the T-800 Terminator's secondary goal is to simply to continue to survive long enough to get further orders from Skynet. We should of course note that this would give the machines a vast technical boost, saving Skynet some 30+ years of R&D into terminator technology.
With conditions like those so far encountered on this mission, Terminator could operate indefinitely, plow through all opposition, and complete the target elimination, then stagger programless through the nuclear devasta tion caused by Skynet and walk up to its machine masters to be programmed anew.