"Has anyone analyzed the variation of use of unusual word order by Yoda over the course of the various series?" Yes.
The frequency of inverted sentences that Yoda uses throughout the original series and prequels was studied by Michael Kaminski, author of The Secret History of Star Wars.
"How much talk like Yoda, Yoda actually does?"
[Quote from the same article, in relation to the same book]
Most notably in the originals, many of Yoda’s unusual speech patterns come from The Empire Strikes Back, specifically on Daegobah when Luke first meets Yoda.
However, it is also clear from The Empire Strikes Back that this isn't strictly the case, the first line we hear Yoda using is grammatically correct:
Still... there's something familiar
about this place. I feel like...
I don't know...
Feel like what?
A simple sentence, but grammatically correct nonetheless. However, as Kaminski realised, Yoda begins to emphasise his 'backwards' speech in order to make himself appear more vunerable.
(pointing a gun at the creature)
Like we're being watched!
Away put your weapon! I mean
you no harm.
This appears to be a pacifist response to the threat presented by Luke's blaster; an attempt to defuse the situation; and the beginning of the charade Yoda plays in order to disguise himself as a harmless old hermit, in order to ascertain whether Luke truly has the makings of a Jedi... Crafty, the Jedi Master is.
This in fact, leads onto your point about:
"Often, Yoda doesn't talk like people caricaturize him"
Yoda is himself playing a role when he [excessively] talks like this, meaning this 'caricature' of him was deliberately crafted by Yoda. While it is certainly a way of speaking he lends himself to naturally, there are noticeable fluctuations in the times at which he uses these patterns, and the times at which he does not.
I am wondering, why are you here?
I'm looking for someone.
Looking? Found someone, you have,
I would say, hmmm?
The little creature laughs.
(Trying to keep from
Help you I can. Yes, mmmm.
I don't think so. I'm looking
for a great warrior.
Ahhh! A great warrior.
(laughs and shakes
Wars not make one great.
Not only does Yoda use it more here when he's ~imparting wisdom~, but this can also be seen in the Prequel Trilogy, too:
It is in these scenes where Yoda is feigning a crazier personality than he normally exhibits that 57% of the lines are inverted. This compares to only 41% of inverted lines in the rest of the movies.
[This quote was also taken from the same aforementioned blogspot article].
Yoda does, arguably, use more ‘backwards’ speech in the prequels, although he does get more scenes and dialogue in the prequels than in the originals, partially due to his greater screen time. There is one scene in The Phantom Menace in which Yoda’s dialogue is fairly grammatically standard:
MACE WINDU : Be mindful of your feelings...
KI-ADI : Your thoughts dwell on your mother.
ANAKIN : I miss her.
YODA : Afraid to lose her..I think.
ANAKIN : (a little angry) What's that got to do with anything?
YODA : Eveything. Fear is the path to the dark side... fear leads to anger... anger leads to hate.. hate leads to suffering.
ANAKIN : (angrily) I am not afraid!
YODA : A Jedi must have the deepest commitment, the most serious mind. I sense much fear in you.
There are a surprising amount of both linguistic and Star Wars speculative articles/books written in relation to this subject, some of which have indeed analyzed the variation of use of unusual word order by Yoda over the course of the various series. If you want to read any of them, I can add more links to these in my answer.