In the past I've called and asked the hotel if they are willing to price match as I much rather want to be a customer of them, rather than Booking.com, and I think only around two times have they denied my request. Probably because those were very large hotel chains and didn't really care about one customer here and there. But smaller hotels have always accepted price marching for me.

Hotel websites aren't typically that great for good prices, nor for the ease of use. But talking to someone on the phone and quoting the booking.com (esp. genius) price can give you a good anchor, and many hotels will at least match it.

Not even once I have seen a higher price on the hotel's website. But I travel mostly to Europe so your experience might be different.

I travelled mostly Europe and SEA and ask directly for everything longer than a night or two. Yes, listed prices on websites are almost always higher, and no I couldn't get a better price than with booking in the majority of cases.

Usually booking maximal a week before arrival. So maybe that's why?

Maybe it depends on where exactly, or what kind of hotels, how late you are with your booking, or the booking genius I have, however I had the exact opposite impression.

I travel mostly Europe and it happens all the time. Also I have a past working in a travel/booking agency and if you are big enough you can get discount prices from hotels because they know you are going to book them X days of rooms over a month. So, they reduce their margin but gets more exposure/promotion by the travel agency.

With flights it's a different thing, but it can happen as well.