If you buy a good pair of headphones they'll last for over a week of frequent use. Mine tell me the battery level every time I turn them on so whenever it drops below like 20% I just leave them on the charger that night. Literally never had them run out on me in 6 years of using it.
Pretty much. Whenever I turn it on it'll immediately search for any paired devices it can find. Usually that means my phone. Never takes more than 2 seconds to connect.
I only sometimes have issues with the connection when it's connected to both my phone and laptop at once. But it's never a big deal. And with just 1 device it's always flawless.
For reference I own a pair of Bose QC35's. But I imagine all headphones around that price range would be of similar quality. Certainly Bose and Sony will.
A good set will come with wireless charging as well. I can charge my headphones on the back of my phone if I get desperate. Even better, each earbud has its own battery, and the case has its own battery, so if I ever get the low battery warning on my case, I just charge it while still using my earbuds without any issues.
My last set of wireless headphones still work, and I used them for 10 years. I've never had a wired set last more than two, usually because the cable gets damaged.