Don't attach the fence to the house. Aside from putting holes in your siding (not a good thing), your fence and your house will most likely move differentially. The attachment could result in a tear of the siding, which would be a bad thing.
Place the fence post close to the house, then run the fencing top and bottom supports up to within an inch or so of the siding (do leave a little gap, for that differential movement). Place the slats on the supports, allowing the slats to get just a bit closer to the siding.
You could even install a flexible membrane, similar to a product used to seal the bottom of a garage door at the floor, between the fencing and siding.
The answer is that it depends on the exact method of pressure treatment (did the chemical penetrate to the center of the wood or not) and the current state of deterioration. The main thing to protect the wood from is water; you want to keep it as dry as possible, by both preventing large amounts of water washing over it, and also small amounts of water remaining in contact long-term. Water washes away the bug-repelling copper compounds used to treat the wood. So, stacking the wood in a sheltered area (like your shed) with some spacing between boards (you can stack them in a loose grid if you have the floor space, or cut the worst pieces up into roughly square pieces you can use as spacers) should allow the wood the best chance.
Now, this is if the wood is in good enough shape to save. You may think so, but if the wood has already been penetrated by WDIs, the protection offered by the PT chemicals is likely past its expiration date. This is normal for fences; they're out in the elements, and unfortunately the copper compounds used are water soluble, which is how they get them into the wood in the first place (kiln-dry it, then immerse it in a bath of copper arsenide and ramp up the pressure to force the chemical-laden water into the grain). If the wood was well-treated, with green all the way to the core, the parts that haven't already been savaged by WDIs should stay good, but if you cut a board in half and see white wood in the center, termites and carpenter ants can eat that board from the inside out. If you see grey-brown all the way in, rot has taken hold and the board is garbage.
One last thing; whatever you do, DO NOT BURN PRESSURE-TREATED LUMBER. Not in your fireplace, not in your wood stove, not in an outdoor fire pit, not in a bonfire, not anywhere. Burning PT lumber, no matter how weathered and degraded, will release toxic, polluting smoke; the primary chemical used in PT lumber is copper arsenide, which oxidizes when burned to produce arsenic. In addition to the normal harmful effects of wood smoke inhalation, the amount of arsenic in 12 ft of 2x6" PT can kill 250 adult humans. A single 6' 1-by fence slat would still have enough to kill you and your family about 10 times over. If you don't breathe it in, it'll dissolve in the moisture in the air and be carried up to the clouds where it will fall as acid rain.
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As the temperature and humidity change, it will tend to warp again. But, if it's securely fastened to the other fence components, that will help to keep it straight. If it was only a tiny bit warped, you're probably just fine using it. But if the warp was a really bad one, it could do interesting things to the fence as it reverts to its old shape.