Also, correct me if I'm wrong but the heavy AA guns appear like they can't point down over the deck. They can only point upwards or parallel to the surface, but not down at the surface.
This assumption is wrong. The US Mark 12 5"/38 caliber dual purpose (surface and aircraft) mount was the primary heavy AA armament facing kamikazes. It was mounted on nearly every US warship. It had a maximum elevation of 85° and could depress 15°.
Even before kamikaze attacks or flying below radar, torpedo bombers had to attack at very low altitude and would often approach even lower to avoid detection only popping up to aim and release their torpedoes. Torpedo bombers were a huge threat and naval AA defenses were prepared to deal with them.
What I want to know is, did any kamikaze ever impact the side or hull of a ship?
Yes. This account of all ships sunk by kamikazes provides several examples.
Hovey (minesweeper) on Jan. 7, 1945 - The Dictionary of American Naval Fighting Ships gives the following account: "At 0450, one plane flying low to the water came in from the starboard quarter passing ahead of Hovey. A few moments later another plane coming from the port beam was put on fire by Chandler. This plane passed very low over Hovey and crashed on the starboard beam.
U.S. freighter Augustus Thomas, anchored in San Pedro Bay, Leyte, is attacked by a Japanese plane. The ship's Armed Guard gunfire sets the aircraft ablaze but the kamikaze presses home his attack, a wing striking the stack of the nearby tug Sonoma (ATO-12) before it crashes the freighter's starboard side.
I'm sure if you poke around you can find plenty more.
There is plenty of footage of kamikaze attacks available to watch. Many attack from a dive, but many attack level or in a shallow dive. What I take away from it is the pilots are lucky to hit the ship at all, precision isn't possible for barely trained pilots under intense fire.
It seems like that would sink the thing much faster.
There's two things to address here. First, the deck of a ship is far more vulnerable. Second, and here I'm getting into informed speculation a bit, a diving attack has a much higher chance of reaching the target.
Why hit the deck and not the hull? The major problem is that a kamikaze flies above the water, not under it like a torpedo. It will hit above the waterline so the hole isn't going to flood. That said, you could get lucky...
USS Wake Island CVE-65
3 April 1945: At 1744, a Japanese single-engine plane missed the port forward corner of the flight deck, exploding in the water abreast the forecastle. Thirty seconds later, a second single-engine plane narrowly missed the bridge structure and struck the water about 10 feet from the hull. This plane exploded after impact, ripping a hole in the ship's side below the waterline, about 45 feet long and about 18 feet from top to bottom and as well as causing many shrapnel holes.
Then there's the problem of penetrating the heavy hull armor. Finally, there's the real sailor's fear: fire.
Battleship and heavy cruiser hulls were designed to withstand large caliber armor piercing shells usually as large as their own guns. The USS Iowa had over a foot of armor designed to stop a 16 inch armor piercing shell weighing 2500 lbs moving at 1000 mph. In contrast, it's deck was "only" 7.5". This was done to save weight. The trade off was considered acceptable as long range plunging fire and aircraft bombs had less penetrating power.
Most other armored ships followed the same pattern, more armor on the sides, less on the decks. USS Hornet had 2.5 to 4" of belt armor, but only 1.5" protecting her hanger deck.
Light cruisers and destroyers were often thinly armored and were vulnerable. They weren't valuable targets in and of themselves, but they would form the picket line of anti-aircraft defenses up to 80km away from the fleet. As such, they were a target to roll back a fleet's AA defenses. As testified by this list of all ships lost by the US Navy and Coast Guard, destroyers got hit hard.
A kamikaze will do some damage just from the impact, shrapnel, and spewing aviation gas around, but it's superficial. A kamikaze normally carried a small 250 or 500kg bomb. For the bomb to be effective, it has to penetrate the armor and explode inside the ship, else the explosion will reflect off the armor and go (mostly) harmlessly out to sea. Aircraft are very frail things. In order to penetrate they need to maximize their speed and minimize their armor. This means a dive against the thinner deck armor.
Once they crash through that deck armor there's nothing to stop that 500kg bomb. It plunges through unarmored decks and eventually explodes somewhere deep in the bowels of the ship causing damage. Worse, there are a lot of things on a WWII US Navy ship that burn really well. Fuel, ammunition, rags, oil, paint, wood, rubber... it's this fire that has the potential to do damage well beyond what a relatively small bomb can do. A raging fire can put a ship out of action for months. That's exactly what the Japanese hoped to do, severely damage ships so they would have to be sent back across the Pacific for extensive, lengthy repairs.
I consider this section to be speculation based on what I know about WWII anti-aircraft tactics.
There's a number of problems with a kamikaze attacking a ship from the side which makes the attack less likely to succeed. First is speed. An aircraft in a dive will go much faster than one flying level. The faster you're going, the harder you are to hit. It also gives the enemy less time to react and you spend less time under the considerable fire a US fleet could pour out.
Next is to consider the enemy's numerous fighter cover. In air combat, altitude is life. Altitude can be converted to speed. Prowling fighters will be patrolling at high altitude waiting to dive and pounce on incoming attackers, gaining speed in the dive. A low flying kamikaze would be easy prey. A high flying and diving one means the fighters cannot gain a speed advantage by diving.
Then consider what happens when your plane inevitably gets shot up. If you're flying low, you crash. If you're flying high in a dive, you continue diving. If the pilot has any control, they might hit the target. Even if the pilot dies, the plane may still impact the target. This was a major problem for AA guns in WWII. Previously it was sufficient to disable the fragile pilot, engine or controls. Now even a very heavily damaged aircraft is a danger. Weight and volume of AA firepower had to increase as well as accuracy.
Flying low over the ocean in a WWII fighter while under fire is a surprisingly difficult task, one that many poorly trained kamikaze pilots would not be able to accomplish. Flying low wouldn't get you under AA, but it might get you under radar or difficult to spot in the haze. Poorly trained pilots were likely to crash into the ocean, get hit by a wave, or fly too high and be detected.
Finally, hitting the side of a ship is surprisingly difficult, especially for a poorly trained pilot under intense fire in a probably damaged aircraft with surface guts of wind blowing them around. As huge as WWII ships seem, there wasn't much hull sticking above the waterline. In trying to hit that narrow target you're likely to hit the ocean, fly over the top, or glance off the deck. In contrast diving on a ship is easier, keep your nose pointed at the target and let gravity do the rest. This is part of why dive bombing was so effective.
That said, most kamikaze pilots were so poorly trained they were only able to conduct a slow, shallow, very vulnerable diving attack.