Such precision gunnery was nearly impossible with WWII fighters. Shooting for the bombs would be a much more difficult target than the engine, cockpit, or wing root (where the wing meets the fuselage). There were largely two types of guns in WWII fighters. The first type was wing-mounted guns that were generally configured to converge at a given distance for maximum firepower. Most American and British designs featured this setup, with machine guns being the primary armament. Planes like the Spitfire and Hurricane for the British as well as the Mustang, Lightning, and most US naval fighters for the US. The second type of gun mount was in line with the pilot either through the propeller hub or through the propeller arc. These guns fired straight ahead and often included a cannon through the propeller hub. Many axis planes featured this alignment, though US planes like the P-38 and P-39 also had this type of arrangement.

And was, say, a B-17's bomb bay as armored as the rest of the plane?

No, there was little armor at all on a B-17 or most other bombers. To attack from below was difficult but possible. Also, most heavy bombers of the time used internal bomb loads--wing-mounted bombs were generally featured on dive bombers and fighters outfitted for ground attack.

Would it be easier to just shoot the plane itself?

Yes, yes it would. In either configuration accurate gunnery was difficult at any kind of range. Even bombers were moving in three dimensions through the sky and were varying their path--to avoid flak if nothing else. So you had to predict where your target would be by the time your bullets or shells reached it, maneuver yourself into position to make the shot, avoid incoming fire from the bomber and its escorts, and provide the right amount of lead for your shots to hit your target. None of these were easy to do. If your shots converged at a point out in front of you due to having wing-mounted guns your fire was much less concentrated at any other distance. You may have been firing six or eight machine guns, but the sky is still a very large place and it becomes an incredibly busy place during combat. Even if your guns fired straight ahead it was still difficult to produce accurate gunnery. Gyro-gunsights were available in WWII in some fighters and they did help, but later improvements like radar-assisted range finding would help even more. Even then, targeting the engines, cockpit, or wing root was much more likely than trying to pinpoint a relatively tiny bomb.

What about rockets?

The rockets mounted on planes in WWII were smaller than the bombs were. Most of them were mere inches in diameter, which would make them even more difficult to target than bombs.

As always, followup questions by OP and others are encouraged.

In the book "General Kenney Reports" George C Kenney says in the summer of 1942, the woefully out numbered Fifth Air Force had a serious problem. Its bomber pilots thought their bomb payloads were extremely vulnerable to exploding if hit with a bullet. So when a Fifth Air Force pilot or air crew saw any Japanese planes in the vicinity, they dropped their bombs right away and headed for home.
Kenney put a stop to this by telling them that bombs don't explode until their fuses activate them. Bombs have to fall several thousand feet before the fuses activate them. The pilots did not believe him, so he had to show them, by having a P-39 shoot at a pile of 500 pound General Purpose bombs, from a distance of 500 feet. The bombs did not explode, but the pilots could see the evidence that fifty caliber and even 37 millimeter shells were hitting and ricocheting off of the bombs. That put an end to the early dropping of bombs, by Fifth Air Force bomber pilots.