Do the events of previous terminator movies influence the intelligence of later terminators?

Question

The Terminator series plays hard and fast with time paradoxes and the like. However the answers to the question "What does a Terminator do once its mission is accomplished?" suggest that in each movie the Terminators get a little bit smarter.

For example, look at the mission parameters of each 'evil' terminator:

Terminator 1:

From: Skynet
To: T-800
Subject: Kill Sarah Connor
   Kill Sarah Connor, she's in LA.
   Maybe just kill all Sarah Connors.
   You are just too dumb to know if it's the right one, so kill them all.

Terminator 2:

From: Skynet
To: T-1000
Subject: Kill John Connor
   Kill John Connor, he's in LA and lives with this foster family.
   Also, the last guy messed up, so Sarah is still there and she is pissed.
   Maybe see if you can find the guy who made me.
   She might want to kill him if she finds out I'm trying to kill her son.

Terminator 3:

From: Skynet
To: T-X
Subject: Kill Mr and Mrs John Connor
Attachments: list_of_resistance_dicks_and_their_locations.xls
   Kill John Connor, we think he's in LA, but can't be sure.
   The last two guys messed up, so he's in hiding, so please try and find him. Oh and kill him.
   But hey if you can't find him that's cool, I've attached a list of his lackeys.
   Just kill them until he shows up.
   PS: Don't worry about Sarah, she's dead LOL. Puny humans, am I right?

As we can see, the mission parameters for each successive Terminator gets smarter with more scope for interpretation.

Is there anything to support the idea that each terminator adds more technology into the timeline that makes building terminators easier, or smarter?

For instance, in Terminator 2, Miles was impressed with the technology of The T-800 from Terminator 1, suggesting that he'd not seen this kind of tech, that made developing further tech even easier, leading to smarter robots sooner, leading to the more intelligent T-800 that saves John in Terminator 2, and so on.

Answer 1

The 1984 CPU and Skynet by predestination

The OP writes:

For instance, in Terminator 2, Miles was impressed with the technology of The T-800 from Terminator 1, suggesting that he'd not seen this kind of tech, that made developing further tech even easier, leading to smarter robots sooner, leading to the more intelligent T-800 that saves John in Terminator 2, and so on.

It is true that the version of Skynet that Miles Dyson was developing would have been based partly on the CPU from the T-800 from The Terminator, and this same instance of Skynet was most likely responsible for the T-1000 sent to 1995 in Terminator 2. However, this could be part of a predestination loop. We already have one such loop: Kyle Reese is John Connor's father, and so John cannot exist unless events occur so that Kyle is sent back to 1984. Similarly, Skynet's existence may depend on the CPU being harvested from the disabled T-800 in 1984. If this is the case, then the existence of all time-travelling robots depend on Miles working with the tech from the T-800 (rather than being the source of an improvement) — more on this later.

T-800 intelligence

Now, I see no evidence that the T-800 from The Terminator was especially "brutish and dim" (to quote OP's comments, which are relevant to the question). I believe that actions required for mission objectives are being confused with level of intelligence / functionality. Its one mission was to kill all Sarah Connors, and there was no reason, at least a priori, for the T-800 to use advanced tactics or behave especially covertly. In fact, it was the intervention of Kyle Reese, a soldier from the future, that led to the T-800's defeat. Also, I see no hard evidence that the "Uncle Bob" T-800 from T2 is more intelligent than the previous T-800. If anything, part of its reprogramming is to be slightly more talkative, as part of its mission is to convince the young John Connor that it can be trusted. We should not confuse silence with stupidity.

Given that the T1 T-800 failed to accomplish its mission objective, Skynet sent a more advanced Terminator — the T-1000 — to 1995 to kill John as a 10-year-old. There is no direct evidence that the technology of the original T-800 contaminating the timeline in 1984 is what led to the T-1000's development. As I said, the T-800's presence in 1984 could have been part of a loop that leads to Skynet itself and therefore becomes an essential event for all Terminators. If anything, before Sarah, John, and "Uncle Bob" interfere with Miles' project in 1995, Cyberdyne seemed on track to initiate its AI project in the very near future. Given that Judgement Day did originally occur in 1997, and given that Miles would have been nowhere near to his level of progress without the future tech (by his own admission), it seems that Skynet's existence does depend on a loop of this form.

Conclusion

In short, the OP's suggestion could be correct, but I can see no evidence for it and can find no confirmation of it in any of the Terminator franchise literature. It is more likely that the introduction of the T-800 into the timeline at 1984 is a pivotal event required for all other Skynet-related events to occur (not just improvements at the level of single Terminator models).

Answer 2

Each T-800 (or T-101, whichever you want to call it) that has been sent back has had largely similar goals: kill one or more individuals to make Skynet's eventual takeover easier.

Let's rewrite your snarky instructions into something more plausible:

Terminator 1:

From: Skynet
To: T-800
Subject: Kill Sarah Connor
   Kill Sarah Connor, future mother of Resistance leader John Connor.
   Subject lives in Los Angeles, exact location unknown.

We don't know what the Terminator would have done if it had succeeded, but we do know it was being methodical. It knew enough about late-20th-century culture to locate a phone book and search for Sarah Connor, and it probably had other sources to use to check for her afterwards (such as breaking into the Health Department and finding birth records, or checking school/university enrollments) in case Sarah Connor wasn't one of the ones listed in the phone book

Terminator 2:

From: Skynet
To: T-1000
Subject: Kill John Connor
   Kill John Connor, future leader of the Resistance.
   He is approximately 12 years old.
   Attached are psychological profiles of Sarah Connor and John Connor, based on surviving records.

The failure of the first Terminator left behind artifacts which improved the next iteration of Skynet, allowing for a cleaner takeover and more surviving records. Basic human psychology hasn't changed, and Terminators are programmed to understand it - given the 'same data records as [Arnold's heroic terminator] has', the T-1000 could easily assume that Sarah would try to destroy Skynet. Most of the rest of the plot of the movie follows simply by using basic psychology.

We see in the second movie that the T-1000 uses the police car's in-car computer system to access records on John Connor, and that he gets a photo from the foster family.

Terminator 3:

From: Skynet
To: T-X
Subject: Kill Mr and Mrs John Connor
Attachments: list_of_targets_and_their_locations.xls
   We have been unable to locate John Connor following the failure of the T-1000 model.  Please find attached a list of secondary targets, and what is known of their pre-Judgement Day activities.  Included in your data files is everything we know about John Connor.  If John Connor is located, he becomes a primary target.

Simply put, the Terminators have standard, pre-loaded information and data on how to integrate into human societies. While I'm sure the parameters have to change in order for them to integrate into pre-JD society, human psychology doesn't. The Terminators can model human behavior fairly well. If you listen in Terminator 1, Kyle Reese doesn't say "early Terminators were unable to fit in as humans", he says, "Early models couldn't pass - rubber skin wasn't convincing."

The basics of target locating are not novel techniques Skynet has to program in for time travel robots, it's part of their basic package. The Terminators parameters don't increase in complexity (although the last one is certainly given more targets than the first was shown to have), but the society in which they are dropped does increase in general technological level. This makes it much easier to locate their targets. I'm sure that the changes made to the past (such as increased computer power due to Terminator-derived enhancements) have changed the way the Terminators are designed or built, but not in any significant way.