Having had an insight recently into the White Crane derivative of Rokkishu, it certainly seems to me that Tensho as we know it (Miyagi lineage) was heavily influenced, if not directly developed from, the Chinese form. The patterns are almost identical, as are the hand techniques. With just a slight difference in emphasis in parts, Tensho seems almost a "poor man's version" of Rokkishu.
As for the animal hands, they're there in both versions: snake (open hand palm down "outer block"), crane (open hand palm up "inner block"), tiger (fingers up "push out"), leopard (fingers down "push out") and dragon (upwards koken - "standing dragon", sidewards koken - "laying dragon"). The characteristics of the animal forms are to be found in the applications of each movement. You also have mawashi uke in there too; as Boz says, this kata really is a combat system in a very condensed form.
As to doing the form under tension, I can see no purpose whatsoever for doing so. Part of the kata is it's chi gung rooting and power sourcing, which it totally lost the second you tense up. The same is true of Sanchin kata - personally I rarely practice "isometric" sanchin, but strive for a soft, flowing sanchin kata with emphasis on connecting the upper and lower body and sourcing power from the feet. Doing either kata [but especially Tensho] "under tension" shows a complete lack of understanding for what the kata contains in my opinion. If I want conditioning, I'll go and lift weights of wave my chishi, bo or sai around.
As to the original questions...
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1. The pattern starts with... a tricky hand moving bit, then a slow push bit, followed by another tricky hand bit and a slow push - then we have the paint-the-fence, paint-the-house bit (up, down, side to side) - how come this bit doesn't follow the pattern appearing in the first part, but the last section (mawashi uke) does?
I need to see the version of the kata you're doing before commenting fully. If it's what I think you're asking, then the answer is "because it does".
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2. Why is it sometimes done under tension and sometimes under none at all?
The former has got the wrong end of the stick and the latter may [or may not] have some clue as to what the kata is about.
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3. Is it a slow or a fast kata?
It should be [IMO] a naturally flowing kata; somewhere inbetween the two.
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4. What is the kata's fundamental purpose?
It is to decide who buys the next round of sake...do you not know your Okinawan pub games?
Oh, you mean in the dojo, not in the pub? Well, what are all kata for? AFAIK, Tensho is a mnemonic for remembering the animal hands and their application (providing you have been taught them), it's a rooting exercise, a chi gung exercise for posture, breathing and "chi cultivation and activation" [if you subscribe to that theory] and it makes the tea in the morning whilst ironing your shirt. Every good home should have a Tensho.

Seriously though, like any kata, it contains a fighting system; how you use it is really up to you and your teacher. What it isn't is a system for developing a six pack and clearing your throat.
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5. What are the kata's origins?
The Chinese form Rokkishu (six hands?), which [may have] originated from or been inspired by the Bubishi and the [6] hand forms. It seems most likely that the White Crane form Rokkishu was inspired in name by the similar hand forms shown in the Bubishi, and that Miyagi was influenced by this form [copied it] in forming Tensho as we know it. Hey, not being 96 years old I wasn't there; Boz may have been, but I wasn't.....
Tom wrote:
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I`ve always been under the (mis)apprehension that Tensho was the counterpart to Sanchin.
I`ve heard abou these "animal" forms before. If they`re in Tensho, whats to say the`re not in the other kata that were derived from Q`uan Fa?
Animal hands are quite prevailant in the Goju kata. Alot of people here will know Saifa. Crane and tiger and leopard are very evident there. Dragon appears in Sanseru and there's a smattering of the little critters throughout the sylabus.
I think I've exhausted my 15 minute expert knowledge on the subject.

It's a kata I'm just starting to get to grips with after a couple of years. A kata that's simple to learn on the surface but has a lifetime of complexity waiting to be explored.