Never, never,
never use soap! If you must, boil some cold water and, using a separate container, add the decorations & plants (if silk), and pour the scalding water over them. I only do this part of the cleaning when his decorations start getting algae or if I'm doing a 100% water change anyway. You'll be amazed how effectively simple boiling water will clean decorations!
Also, to reduce stress, you could probably just do partial water changes with your betta in the tank. Darby stays in his tank for anything other than 100% water changes, and seems more curious than anything about his turkey baster. I hate having to catch him with the net, and he certainly doesn't appreciate it!
Another tip I've learned for keeping new water near the tank's temperature is to boil small amounts of it, then add it to the aged water- BEFORE adding it to the tank, of course. I test with my fingers to feel if its too hot or too cold compared to the original tank water. (It's often too cold even when I leave it right next to the tank.)
Also, even though I use aged water, I make sure to add dechlorinated to the water as well, just as an added protection measure. Chlorine is TERRIBLE for fish, so better safe than sorry.
So to answer your question about cleaning with chlorinated water, I would say yes, add some more dechlorinator after everything is all spiffed up.
Also, someone suggested getting a stick on the outside thermometer, but I've found that in my 10g tank, my internal hanging thermometer is more accurate, just FYI.
Glad to hear you got yourself a cover for the jar, and your little guy is just beautiful. I wouldn't worry about the gray on his face, it's likely just his coloring. Lots of bettas have different colors on their faces- My Darby including.