If I rent a bike for sightseeing, is there any parking space to leave it near temples, shrines, markets, etc. (Kinkakuji, Fushimi Inari, Nishiki, Gion...)? Is it safe?
A simple Google search for "京都 駐輪場" (Kyoto bike parking lots) brought me to this page, which contains all city government-approved bike parking lots. Most of them are cheap (150-300 yen per day), and many of them offer 3 hour free parking. But unfortunately, this page is in Japanese, and the English language version of this website doesn't contain a lot of info about parking.
But then you could always find your destination on Google Maps, click "nearby", and search for "bike parking". If you see 自転車駐車場 or 駐輪場, then it's a bike parking place. There's also an English language site here if you're too lazy to Google.
As for safety, I believe Kyoto is a rather safe place. Of course, you should bring your chain lock or U-lock, and lock up your bike. Do NOT park on the streets as it is illegal in Kyoto; your bike will be towed away. Not a pleasant situation.