Quality is what matters!
Terrible anchovies (any you get a fast food pizza place) make terrible pizza. Good anchovies (white usually) with the proper seasonings, great pizza. Has to be on a crispy, thin crust though. Something about thicker crust or even brooklyn style "floppy" thin just doesn't work with anchovies for me. Overall though, definitely better for a Caeser.
Re: Mushrooms. Mushrooms are a classic topping, and one of the greatest foods of all time, how dare people denigrate their good name!

But, again, it's quality. Random canned mushrooms or low end white mushrooms, meh at best, bordering on outright gross. Some nice portobella or shiitake mushrooms, soooooooo good. Have to get a proper sear on them in a good brick oven or under a broiler though, you don't want them soggy. Goes particularly well with a good pepperoni. Alternatively, sautee them first and get some of the moisture out.
Hawaiian pizza is a monstrosity. Pineapple should not be served hot, ever. Even grilled pineapple should be allowed to cool.
Crust depends on the toppings. Brooklyn is the best all purpose. Not a fan of the chicago style deep dish, it's just a wannabe pot pie, but works with some toppings/fillings like spinach and feta.
BBQ on pizza I don't care for. I've had decent BBQ chicken pizzas, but they might as well have been any random sandwich. Waste of pizza calories.
My perfect pizza is simple: Brooklyn style crust with a classic red sauce, Mozz + Prov + Cheddar blend, tons of pepperoni, and extremely finely ground beef (EXTREMELY finely ground) placed on top of the pepperonis so that as the oil from the pepperoni cooks out it fries the beef, with some roasted garlic for extra kick. Top with red pepper flakes and the fakest, cheapest, grated parm there is.